When they had the Shipment 24/7 level I was able to level up one of my SMG weapons, but that game type doesn't seem to be available any more. Is there another way to level up weapons quickly?
The only way to gain weapon XP is to kill enemies with your firearm. Because of this, you need to need to play on a map which has a lot of combat so you can kill people quickly.
With Shipment 24/7 and Shoot the Ship gone your best bet is Shoot House 24/7. It's the next smallest map and you can guarantee to play it every match. Other, large, Domination 20 player maps also offer opportunity to kill lots of opponents if you're good at hunting enemies down, but the large levels might not be balanced against your short-ranged SMG.
In any case, you'll also want to use a Double Weapon XP token. As the name implies, it doubles the rate that you earn weapon XP.
These don't stack so only use one at a time. There's a variety of ways to acquire these tokens, including completing challenges, campaign missions, or using companion apps. It can be accessed and consumed from the game lobby by pressing 3 on PC or clicking in the right thumbstick on consoles.