I closed on my first rental today and we are going to be living in one of the units. 

The unit we are moving into currently has tenants that I talked to today and will be moving out at the end of July. From the seller, we got credited the tenant's security deposit, prorated rent for June, and last month rent (which will cover July). 

Do I need them to sign a new lease for June and the rest of the July or will a 39-Day notice be enough since I already have rent payments and the security deposit? 

I don't know Idaho law, but generally, upon the completion of a 12 month lease, a lease automatically becomes month to month, unless the lease states otherwise. Most states only require a 30 day notice of non-renewal for a month to montheass, but again, you would have to check, as I am not familiar with Idaho law

What does their existing (inherited) lease say? Assuming it ends in July or that they are month to month already, 30-day written notice is fine. 

http://www.ag.idaho.gov/publications/consumer/LandlordTenant.pdf

Just to be on the safe side and protect yourself, I would provide the notice of non-renewal. You never know what could happen, that way if necessary you can start the eviction process right away if they change their mind. Congrats on doing a house hack!

Hopefully you have a copy of the inherited lease, and will go by that. Otherwise, I would absolutely give a written notice of the termination of their residency. The sooner the better. 39 days might not be enough. My lease specifically says 60 days from tenant or landlord.