Yes, an astonishing number of gun owners support storage and lock regulations, and support background checks.
The opposition uses this fact to label certain practices as something other than what they are. For instance, you can create a 'mandatory waiting period' by saying all gun transfers need a background check and then making sure that check takes a month.
I'd love for there to be a truly instant background check, one that takes 90 seconds or less, and one that isn't legally required but the use of it makes the seller legally immune from criminal or civil lawsuit due to the wrong person getting a gun that once was in the seller's control.