I would like to hire a nurse to help with my father-in-law's laryngectomy stoma. Is this possible?

You need to talk to the RN in charge and see what she can do. If the facility can do it, his cost of care will go up. Talk to his PCP about in-home care that Medicare will pay for. There is a member who was able to get "intermittant" care for an Aunt and her mother with Medicare paying.

Check into an agency that specifically supplies LPNs and RN nurses. One agency that came recommended to us was brightstar care

If he's already in a facility, this should be something that is dealt with as it is needed. You don't need to hire someone to come in simply to wipe the area and keep it clear of mucus. What would this CG do the other 23-1/2 hrs of the day?

Yes, it is possible.

Not from this forum though. You need to contact a local agency or go speak with his doctor's nurse and find out if they know anyone looking to make extra money.