Lots of confusion in the area.
In general, if you are a non-resident (not a US citizen nor a permanent resident = green card), federal law makes things hard. Here's the relevant law: 18USC922(g)(5): It shall be unlawful for any person ... who, being an alien (A) is illegally or unlawfully in the United States; or (B) ... has been admitted to the United States under a nonimmigrant visa, ... to ... possess ..., any firearm or ammunition. The really important part is the word "possess" here. It does not mean to own, it means to hold in your hand or to control. So if you are prohibited, you are not just prohibited from owning or receiving a gun, you can't hold it. Not even in a training class. So far, so bad.
You are on an H class visa. That is a non-immigrant visa. Done, you're out. This sounds like awful news. Note that an I-94 does not help at all; it is about the non-immigrant visa status.
BUT: The important part is what I left out, namely the first "..." after the letter (B), which is exceptions listed in 922

(2): Subsections ... do not apply to any alien who has been lawfully admitted to the United States under a nonimmigrant visa, if that alien is ... in possession of a hunting license or permit lawfully issued in the United States.
So here is what you have to do before you can legally use guns: Get a hunting license. Which is not trivial; last I looked, it was a weekend-long class, and it takes several hundred $. Note that there is not a catch 22: to get a hunting license, you do not have to use guns to demonstrate proficiency with guns first. And once you have a hunting license, the exemption applies to everything, rifles, shotguns, pistols, even just plinking and target practice.
Now, is this restriction really enforced? Some ranges enforce it; others are happy to check your driver's license and rent you a gun and sell you ammo. If you were to go shooting with friends at their backyard range (borrowing their guns), and a sheriff came by, they would probably not blink an eye, in particular since you speak good English (perhaps with an amusing Canadian accent, eh). Local (state) law enforcement typically doesn't see it as their job to enforce federal law. But it is still a massive risk, since if you DO get hooked up, it's a federal felony, and you can kiss gun rights in the US good bye for life (and your visa probably too). So step 1: GET A HUNTING LICENSE. Yes, I'm meaning to shout.
Once you have that, take whatever training class you want. Go to a range and rent guns. They may still refuse you (because they are dumb), but legally you can. Buy yourself a gun, or 10 or 20. Shoot as many rounds as your paycheck allows, and stockpile tons of ammo in your tiny Bay Area apartment. Yes, you can own guns once you have a hunting license, just like gringos and green card people can.
And since I'm such a Debbie Downer today, a few more warnings. Do not let your hunting license expire if you own a gun! Renew it religiously before the expiration date. Otherwise, instant felon. And the nasty warning: A few months ago, some court (I think in Arizona) said that obtaining a hunting license just for the shooting hobby without any intent to hunt does not fulfill the requirements of the law. I think that judge is blatantly wrong (what part of the black letter of the law don't they understand?), but that's out there. Fortunately, you can relatively easily hunt. Come to my house, I'll lend you a rifle suitable for Coyote (a 223), and let you stomp around in our back 40 for a few hours. You might actually see a coyote (they are not that rare). If you do, feel free to aim and shoot. You will probably miss. Should you hit (by mistake?), we'll have to spend half hour burying the carcass (a tractor will make quick work of that); that would be disgusting. Coyote season is all year long, and 223 is a fine caliber for them. Just because you are a bad hunter doesn't mean you didn't intend to hunt. That little performance should be enough to shut up that silly judge or any imitators.
Hope this helps!