The vast majority of modern adhesives are synthetic. Modern synthetic adhesives, such as epoxy, cyanoacrylate, silicone, polyurethane, and acrylic, are far superior to early adhesives made with animal collagen, dairy casein, plant starches, and tree sap. Synthetic adhesives are made by chemically-reacting base materials derived from petroleum and natural gas - example:
Polyurethane - Wikipedia
It is commonly believed that "white glue" (such as Elmer's glue) is made from animal material. The cartoon-drawing of a cow on the Elmer's glue bottle is perceived as proof of this. However, it's not true. Common white glue is polyvinyl acetate (
Polyvinyl acetate - Wikipedia ), a synthetic adhesive made by the polymerization of vinyl acetate, a monomer made from ethylene (derived from natural gas) and acetic acid (derived from petroleum-synthesized methanol, or from fermentation of plant sugars).
Animal-content adhesives still have limited use in certain low-quality applications, such as binding telephone books and junk mail. Regular books, however, are glued together with synthetic adhesives:
Bookbinding Adhesives . Telephone books, of course, have largely been made obsolete by the internet.
Very small amounts of animal material are used in the processing of some plastics. For example, calcium stearate (made with stearic acid, which has historically been made from beef tallow:
Stearic acid - Wikipedia) can be added to plastics as an acid neutralizer, processing lubricant, and release agent. The amount of added animal material is tiny; typically less than 1000 ppm, or 0.1%, of stearate:
Calcium stearate - Wikipedia . Even this is being phased out for certain products, due to concerns over bovine spongiform encephalopathy ("mad cow disease"):
https://www.nordsonmedical.com/newsletter/animal-free.htm .
Lots more information is available on this, although the topic is mostly academic, since modern adhesives (including hardware store glues) are all synthetic.
If you are interested in helping to develop alternatives to other animal products, you might pursue a career in chemistry, biology, or food science.