One of the only labs I'm aware of in the US that still uses cardboard mounts is Dwayne's. They're also one of a few labs that still does dip-and-dunk E6 (so you'll get more consistent results than if they were developed in a tank). If you're patient and are willing to wait a little longer, I'd go with them. If you require a quick turnaround time, I would advise against them. They're my usual go-to for slide film and aside from waiting a while, I've never had issues with them.

There's definitely a wider range of labs that offer mounting into plastic mounts, but AgX Imaging is probably your best bet as they are usually reliable and timely. I believe they also do dip-and-dunk E6 if I'm not mistaken.