I recommend sending samples ASAP.
Do you have a good Extension service there (or Bureau of Forestry or Department of Agriculture???)? They probably would want to know if Oak Wilt is in a new area, so they may help with samples.
If the lab only runs cultures, it can be difficult to get them a good sample. Ideally it would get there the same day you cut it. Otherwise figure how late in the day you can get it to USPS/UPS/FedEx and expect next day delivery. Cut the samples at the end of the day and get them shipped as fresh as possible. Apparently the pathogen is hard to grow and dies quickly once removed from the tree so its easy to get a false negative...which is crazy for how aggressively it spreads in the landscape.
Or, send to a lab that uses DNA testing. Not sure if Penn State does or not...worth asking. I send samples to Research Associates Laboratory. (I'm guessing they started in the veterinary field because their website is vetdna.com) Horticulture form:
https://www.vetdna.com/application/forms/horticulturesubmissionform.pdf for $20 might even be worth a second sample even if you send samples to another lab that cultures for the test (which is really the "gold standard" for diagnosing). You can send completely dead branches...just make sure they were likely killed by whatever is going on. I'm guessing they can tests leaves too? Maybe call and ask that. I've only sent them bark samples from the trunk off of suspect pressure plates on trees that were already dead.
Tricky part: We don't want any pruning on oak trees this time of year. I'd paint the wounds as soon as the cuts are made.
Next questions:
Are you familiar with oak wilt management (on a landscape scale)? Are there other oaks within root-grafting range? That'd be the big reason to jump in it now...you have to go after it
hard or it keeps spreading. "let's not remove this tree that's 8' away because its asymptomatic", or "root graft cutting is too expensive" are recipes for losing more trees. If its just one more tree that doesn't spread into a larger population, might be worth cutting root grafts and treating instead of removing - as long as they understand it may already be infected but not showing signs.