Bills have already been introduced and are in committees:
HR 127 To provide for the licensing of firearm and ammunition possession and the registration of firearms, and to prohibit the possession of certain ammunition.
https://www.congress.gov/bill/117th-congress/house-bill/127/text?r=109&s=7
HR 130 To require the safe storage of firearms and ammunition, and to require the investigation of reports of improper storage of firearms or ammunition.
https://www.congress.gov/bill/117th-congress/house-bill/130/text?r=112&s=7
HR 135 To require the Director of the Federal Bureau of Investigation to report to the Congress semiannually on the number of firearms transfers resulting from the failure to complete a background check within 3 business days, and the procedures followed after it is discovered that a firearm transfer has been made to a transferee who is ineligible to receive a firearm.
https://www.congress.gov/bill/117th-congress/house-bill/135/text?r=117&s=7
HR 137 To authorize funding to increase access to mental health care treatment to reduce gun violence.
https://www.congress.gov/bill/117th-congress/house-bill/137/text?r=119&s=7
HR 167 To prohibit the transfer of a firearm at a gun show by a person who is not a federally licensed firearms dealer.
https://www.congress.gov/bill/117th-congress/house-bill/167/text?r=149&s=7
An unrelated but interesting Bill, cosponsored by 72 Democrats:
HR 262 To prohibit the imposition of the death penalty for any violation of Federal law, and for other purposes.
https://www.congress.gov/bill/117th-congress/house-bill/262/text?r=244&s=10