Caveat: I'm not familiar with WV landlord/tenant law, but these are general guidelines.
The seller should be turning over the S8 tenant's SD to you during closing. Check your HUD statement and make sure it is included. If it isn't, I guess you could try to fix that in the 11th hour, but ultimately you will be one responsible for returning it to that tenant for whenever they move out whether you get it from the seller or not.
You can create new leases with your all's information as the property owner, but the terms need to remain the same until the end of the current leases. If the current lease(s) are month-to-month, then you can change the terms. Usually with a 30-Day Notice, but your area might require longer.
Assuming the no security deposit tenant is already a month-to-month or will be by the time 3 months are up and you raise the rent, I'd suggest keeping their new lease a month-to-month. Paying the SD off AND higher rent might be more then they can chew, even if they lie to you and/or themselves that they can do it. Then you can just give them a 30-Day Notice to Vacate if they fall behind in rent. In the vast majority of places...including my sleazy New Orleans...even if you can't evict someone for non-payment of rent, you can evict them for being a holdover after overstaying their lease.
After closing, post a letter on their door and/or contact the tenants if you have their phone numbers, and arrange a mutually good day/time for them to sign the lease with you all. When you meet with the no SD tenant, I'd give them a heads up that their rent will be staying the same for the next three months but, after that, it will be raised to $X and they will need to pay the SD in a payment plan over Y months. I'd be sympathetic. Let them know you realize that might make it difficult for them to stay, but they have time to decide what to do. If they choose not to stay, then they can give you a 30-Day notice of their move-out date. Since they'll be signing the new lease anyway, I would include the date and amounts that the rent will change to X and the SD payment plan for Y goes into effect.
Last, but not least, hopefully you have already looked into the process for S8 paying that tenant's rent to you. But unless things have changed mightily from when I was last in a similar situation, I don't picture S8 being able to do it as fast as you closing today and getting a direct deposit from them on Monday (first business day after the 1st). I personally would have pushed out closing to Monday. The seller can get paid like they normally do by S8 and then the pro-rated rent would be turned over at closing. Giving almost a month to get set up as a S8 property owner (if needed) and that tenant switched over to you.