You should be fine, folks take an out-of-state permit to training all the time - just let the recruiters know this so they can double check. At the end of the instructional phase, you'll test with a MODOT examiner, and will receive a Missouri CDL. Dispatch will route you and your trainer through your home ASAP so you can do what's necessary to transfer the CDL to your home state.
I got my Tennessee learner's permit prior to starting at Springfield & that works just fine. A Missouri permit is not necessary unless you show up without any state's permit. So you're good to go with your Delaware permit. I don't know how the endorsements work with a permit. I'm pretty sure a hazmat endorsement will not transfer from state to state when you xfer your actual CDL back to your home state. Robert