Actually, I don't think so. There's a good discussion of how the origination clause is interpreted
here. According to that article:
...[T]he House includes within the definition of revenue legislation not only direct changes in the tax code, but also any fees paid to the government that are not payments for a specific service, and any change in import restrictions, because of the potential impact on tariff revenues. <snip> The Supreme Court has occasionally ruled on origination clause matters, adopting a definition of revenue bills that is based on two central principles that tend to narrow its application to fewer classes of legislation than the House: (1) raising money must be the primary purpose of the measure, rather than an incidental effect; and (2) the resulting funds must be for the expenses or obligations of the government generally, rather than a single, specific purpose.
I take this to mean that the origination clause applies to bills intended to
raise revenue. So none of this is relevant, as the current legislation proposes grants to the states for the specific purpose of funding the implementation of the new background check requirements, but says nothing about how these grants should be funded.
It certainly isn't relevant to fees collected by FFLs, since those don't go to the government at all.
None of this is to say that Rep. Stockman won't try to do this, but it's not likely to do much. And since the Senate bill is very unlikely to get anywhere in the House, it looks to me more like a bit of grandstanding on his part.