Laws requiring residents or citizens to own weapons for militia service were common in colonial times, and I believe in early post-colonial times, under the Constitution. Such laws had been in effect in England for hundreds of years. Such laws often required that the weapons be maintained and in good working order.
I see nothing obviously unconstitutional in the proposed law. It does not mandate a particular manufacturer as currently proposed, but even if it did, a state is allowed to favor a manufacturer in making purchases if it so chooses and if its own law permits.
A person with a genuine religious objection to owning firearms might be able to sue to avoid the application of the law to that person, if the law as eventually passed did not provide such an exemption. If so, that would not make the law generally invalid.
The law would not exempt people from Federal background checks. If a person was unable to pass such a check, and thus could not legally purchase such a weapon at retail, such a person might be able to prevent enforcement of the law on him- or herself, but that would not make it generally invalid.
In any case the law as proposed provides no penalty for failure to comply. But siuch a provision could be added.
How likely such a law would be to pass, and to be enforced, is a matter of politics, not law. So is the question of how wise such a law would be.