Washington has state-level preemption for gun laws. It's highly unlikely this law would survive a legal challenge on that basis, so arguments over its merit are probably moot.
Moreover, Seattle's city limits encompass only a fairly small part of the (enormous) Seattle metropolitan area. Only a rather small percentage of the area's gun shops are within Seattle city limits. It would take most Seattle gun owners a matter of only a few minutes to drive to a shop without the extra charges (even given Seattle's truly horrific traffic congestion). I'm not sure what this law's backers intended that it accomplish, really.
Personally, I wouldn't object to paying such a modest amount, despite the fact that like most shooters, my firearms will never be involved in incurring the cost to society of gun-related violence. Of course, that's assuming the money was actually going to be allocated to something that will do some good. Facilitating expanded background checks, funding gang outreach programs, what-have-you...
People who drive on US highways pay a fuel tax to maintain those highways, why isn't it appropriate for people who shoot to pay a tax to cover the cost to the state of that activity?
I think it's a stretch to consider, for example, my target shooting competition and practice as having anything whatsoever to do with the cost to society of criminal use of firearms. The only commonality is firearms themselves, and I think that's far too general a connection. Criminals seldom use rifles of any kind in their crimes, and high-end target rifles simply aren't seen in crime stats (handguns are the overwhelming choice). My gun use costs the state nothing. My car use does.