Laundry drain smells like sewer after washing clothes
How can I determine if the sewer smell is from a lack of p-trap in the wall (ideally without removing drywall)?

It would be very unlikely for there not to be a trap; they are required by code. More likely you have a trap that is being siphoned dry, which can be caused by various problems including poor venting, clogged or partially clogged drain, poor design.

Could the sewer smell come from another clog?

YES... well not really coming from another clog, but caused by a clog or partial clog.

could I just put a p-trap outside of the wall attached to the plate to solve the smell?

NO you can't piggy-back traps. If there is a trap in the wall (and I would bet $5 that there is) you can't install an additional trap.
I would recommend that you ensure that all house drains are running free and clear (by whatever means necessary) and same with all vents. The fact that you merely wet-vacced the line and did not snake the drain line (