That appears to be a Price Pfister tub/shower valve.

Note 1- there are wayyyy more manufacturers/brands than just P.P. and Gerber. And each manufacturer has many (many many) different styles/models/types.

Note 2- it is not just the "stem" that the manufacturer/brand (and style/model/type) applies to; the whole valve assembly is manufacturer/brand-specific.

Note 3- while stems and trim are generally manufacturer/brand-specific and not interchangeable, this particular brand and model is pretty common (at least in the U.S.) and replacement parts are widely available from both the original maker (P.P.) and from others.

Note 4- I highly recommend that you remove the stems and bring them with you when you go to the plumbing shop to get the parts you need.

How many splines are on the broach, how long is it and what's its diameter?

As according to faucetpartsplus.com, that's all you need to get started. They have an entire list for spline counts, listing them (below) to specific manufacturers. If you can't figure it out, they suggest sending them a photo, as they're very interested in selling you parts; manufacturers and their employes might not be so inclined. It's more than likely they'd only be able to refer you to who makes or has the replacement parts these days; not their prerogative.

Even though the link goes to an excellent array of pictures and contains most of the leading brands, I implore you to contact your local parts suppliers who are usually more than willing to help you.

You mentioned plumbing suppliers. Those are places you buy pipe and fancy toilets, you need a repair part supply house.

4 point square broach - Milwaukee, Royal Brass, Speakman, Sterling, American standard, Chicago, Concinnity, Gerber, Price Pfister, Symmons, Zurn
8 point broach - Briggs
12 point .335" broach - Crane Dialeze, Michigan Brass, Wolverine.
12 point .375" broach - Arrowhead, Artistic Brass, Harcraft, Glauber, Price Pfister, Scoville, Universal Brass.
12 point .39" broach - Bradley, Elkay, Fisher, Sears, Universal Rundle.
12 point .415" broach - Symmons.
12 point .485" broach - Crane, Symmons.
15 point broach - Santec, Savoy, Wolverine.
16 point .360" broach - Gerber, Sayco.
16 point .370" broach - Acme, American Standard Cadet & Colony, Barnes, Burlington, Central Brass, Concinnity, Eljer, Glauber, Harden, Kohler Trand, Milwaukee, Newport, Phylrich, Royal Brass, Scoville.
16 point .40" broach - Sterling.
17 point broach - American Brass, Mansfield, Midcor, Phoenix, Streamway.
18 point broach - Briggs, Indiana Brass, Union Brass.
20 point .285" broach - Broadway, Concinnity, Danfoss, Dornbracht, Eljer, Grohe, Jado, Kohler, Milwaukee, Paul.
20 point .415" broach - Broadway, Speakman, T & S Brass.
22 point .375" broach - American Standard old 3/8".
22 point .438" broach - American Standard current 7/16".
38 point broach - Import, Pegasus, Glacier Bay.
D Broach - Delta, Milwaukee, Universal Rundle, Valley.
Oval Broach - Delta, Moen, Mixet.
faucetpartsplus.com