Elmer's apparently doesn't reveal its glue stick contents. Elmer's has at least four different glue sticks without giving the difference in composition. You can find glue sticks that specify use for 3D printers. They contain PVP. Many wood glues have PVA. PVA may be labeled also as PVAc, or as "VA/crotonates/vinyl neodecanoate copolymer" (for example in hairsprays).
It looks like you will need to search patents to find contents of glue sticks. Her are examples.
According to Elmer's patent 6268413, Wood Glue Sticks have PVA, but I haven't seen such a thing:
This Elmer's patent has PVA: High-strength adhesive paste and dispenser
Patent number: 6268413
Abstract: The present invention includes a high strength adhesive and includes a dispenser containing a high strength adhesive. The invention is an adhesive composition comprising: (a) water present in an amount of from about 40 percent to about 70 percent by weight; (b) polyvinyl acetate present in an amount of about 15 percent to about 35 percent by weight; (c) dextrin present in an amount from about 5 percent to about 35 percent by weight; and (d) starch present in an amount from about 0 percent to about 5 percent by weight; the adhesive having at least 30% by weight solids, having a paste-like consistency, and having a bond strength greater than about 2000 p.s.i. as measured by ASTM D-905. The adhesive of the present invention has a very workable consistency that is very much like room-temperature butter or shortening used in cooking.
Elmer's patent with PVP: HIGH-STRENGTH GLUE STICK FORMULATION, Publication number: 20110301260, includes water; a defoamer; a bactericide; an oil; polyvinylpyrrolidone; a plurality of polyurethane dispersions; dipropylene glycol n-butyl ether; sodium stearate; casein; sodium hydroxide; and at least one adhesion promoter.
Another Elmer's patent has PVP: Adhesive applicator crayon
Patent number: 6066689
Abstract: Adhesive compositions exhibiting fast cure time in solid form, e.g. in the form of an applicator crayon or glue stick. The adhesive exhibiting polymer of the composition is based on a mixture of critical amounts of high molecular weight polyvinylpyrrolidone polymer, with a lower molecular weight polyvinylpyrrolidone polymer, with water, an organic solvent, and a salt of a long-claim fatty acid, e.g. sodium stearate. Color indicators and other optional ingredients, e.g. mildewcide, can be included.