(Question: "How did the early chemists make a connection between gram formula weight with 1 mole and Avogadro's number?")
The article cited in the question has unfortunately led to a misconception, because that cited article starts out from a fundamental mistake about Avogadro's work. Avogadro did not discuss anything like what is now (honorifically) called Avogadro's number, he did not deal in absolute numbers or quantities. His important advance was concerned with relative combining proportions and not with absolute measures. Even the title of his original work already makes that quite clear, thus:
"Essai d'une manière de déterminer les masses relatives des
molécules élémentaires des corps, et les proportions selon lesquelles
elles entrent dans ces combinaisons; par A. Avogadro"; Journal de
Physique 73, 58-78 (1811): (emphasis added)
This title becomes in English:
"Essay on a manner of determining the relative masses of the
elementary molecules of bodies, and the proportions in which they
enter into these compounds" (again, emphasis added).
This title comes from a very useful English translation, of which the main parts are given here, of Avogadro's original paper, made in about 1890 by the Alembic Club (Edinburgh). Their translation is also provided with a large number of helpful notes which should assist in the avoidance of misunderstanding .
The notes to the Alembic Club translation also mention the honorific origin of the expression 'Avogadro number', which was named a century after the time of Avogadro himself, after breakthrough experiments by Millikan to measure the charge on single electrons, thus enabling determination of scale factors between macroscopic and atomic-scale measures. Previous to that experimental achievement, the idea of this number had remained an indeterminate concept that there was a definite number of particles in a standard gram-atomic weight quantity (or a different definite number for any other adopted macroscopic scale-unit), but no way to measure any of these supposed numbers or even to verify that the concept was justified in reality.
This is well-expressed in the wikipedia article on Avogadro's constant, thus:
"Accurate determinations of the Avogadro constant require the
measurement of a single quantity on both the atomic and macroscopic
scales using the same unit of measurement. This became possible for
the first time when American physicist Robert Millikan measured the
charge on an electron in 1910."
In other words, the Avogadro number is hardly a constant, but rather a calibration factor, and it's also important that the above statement in Wikipedia doesn't specify which quantity is measured nor on which macroscopic scale. Clearly, different values for the calibration factor
result from different choices of atomic-weight scale and macroscopic mass-scale, although in practice until recently the uncertainties of measurement practically swamped any differences due to variants in
the chosen atomic-weight scale.
So, the answer to the question as put is that the early chemists did not make a connection between gram formula weight with 1 mole and Avogadro's number, @conifold's comment is correct, the concepts named in the question were altogether later concepts, and the earlier chemists dealt with relative measures not absolute numbers or quantities.
The Avogadro number, thus named much later 'after Avogadro', as an honorific not as something he actually dealt with himself, is thus not a fundamental constant of nature, it provides a scale factor to relate two chosen standardized scales, one on an atomic level and the other on a macroscopic level but both of them essentially arbitrary. The present-day value of the Avogadro number is effectively the calibration factor between the current relative standard scale of atomic weights, and the current standard scale that includes the gram, i.e. the number of atoms or molecules in the quantity of a pure compound which is its atomic or molecular weight taken to be expressed in grams.
It may be helpful to add mention of the main historical 'ingredients', so to speak, of the modern concept, they should really include all of the elements of the history of relative atomic-weight determinations:
-- Dalton's (chemical) atomic theory;
-- Gay-Lussac's law of combining volumes;
-- Avogadro's law (initially hypothesis) about the equality of molecular numbers in the same volumes of different gaseous compounds and elements under standardized temperature and pressure -- to begin with.
Then these principles had to be augmented, by further experimental facts that led to the concepts of
-- valency and
-- alternative oxidation states for many chemical elements, and thus to the need for a distinction between (gram-)atomic weight and
-- equivalent weight. (One and the same element, with a single (gram-)atomic weight, would have a different equivalent weight for each of any different oxidation-states or valencies with which it enters into its different compounds: its equivalent weights are related by small-whole-number ratios or fractions with its (gram-)atomic weight.)
The modern value(s) of Avogadro's number thus rests on two kinds of arbitrary decision:
(1) to fix the relative scale of atomic weights, based thus far on ratios, by adopting a convention-based number for the weight of one of the elements in the network of ratios: and then this determines the scale weights of the others upon the adopted convention.
(Among the convention-based numbers (scales) that have been historically used are H=1 (Dalton and more general use until near the end of the 19th century); (chemical) O=100 (Berzelius, used for a time in the earlier 19th-century, thanks to @m-farooq for his reminder of this one); and a (chemical) O=16 scale adopted near the end of the nineteenth century. There were slight adjustments made to the atomic weight scale and its basis during the course of the 20th century which have only very slightly altered the numbers, much less than the change from H=1 to O=16, but the value of the Avogadro number depends on these alterations too.) (For a reference-source, see IUPAC's "History of the recommended atomic-weight values from 1882 to 1997", from 'Pure & Applied Chemistry', Vol. 70, No. 1, pp.237-257 (1998), which also discusses the differences between current and older values and the uncertainties of earlier ones).
(2) A second arbitrary element in the definition of the present-day Avogadro number and mole is to assign to each arbitrary unit of the relative atomic weight scale an arbitrary unit of macroscopic weight or mass. For the purposes of chemical science, that unit has been traditionally and arbitrarily the standard gram, the definition of which has experienced slight alterations too. The practical value of such an assignment is to bring the abstracted relative atomic weights back into the concrete world by matching them up with actual measurable weights.
The chemical terminology used to include expressions such as 'gram-atomic weight', 'gram-molecular weight', and 'gram-equivalent weight' or even 'gram-formula weight', mostly superseded now by the expression 'mole' which means the same as 'gram-molecular weight' i.e. molecular weight on the arbitrary standard scale, but taken to be expressed in grams.
Only after these two steps have been definitely taken and standardised can it make sense to speak of a specific number as the Avogadro number. The Avogadro number can thus be seen to depend on two standardized arbitraries: the scale-unit assigned for the relative atomic weight scale, and the standard mass-measure to be associated with that scale-unit.