Biden to sign executive order to require background checks on more gun sales

They don't enforce the laws they have...and this still won't stop criminals

The answer to that is "Yes" (in this instance) - he can sign an Executive Order regarding aspects about a Constitutional Right. He can't "sign them away", but if there is an Executive Branch policy that intersects with those rights, sure he can (and it can be challenged if it is overly broad or restrictive).

I think the better question is if the President can redefine the definition of a term that is codified in law through an Executive Order. The answer to that is no.

Right now the federal law* defines a 'firearms dealer' as one that is "engaged in the business” of dealing in firearms. Furthermore, "engaged in the business" is defined in law as “[A] person who devotes time, attention, and labor to dealing in firearms as a regular course of trade or business to predominantly earn a profit through the repetitive purchase and resale of firearms, but such term shall not include a person who makes occasional sales, exchanges, or purchases of firearms for the enhancement of a personal collection or for a hobby, or who sells all or part of his personal collection of firearms.”

By contrast, a so-called “private seller” (one who is not “engaged in the business”) is exempt from federal licensing requirements (i.e., conducting background checks).

He can tighten up the view the ATF and other federal law enforcement has regarding "engaged in the business" to catch those that SHOULD be considered in that category (i.e., the ones that pose as collectors or hobbyists but are really regularly trafficking firearms), but he can't unilaterally change the definition.

If he goes beyond the bound of "reasonable interpretation of the law" then he'll be slapped down.
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* Gun Control Act of 1968

They must like getting slapped down by the courts.