On the commissary question first: InmateAid does not handle commissary or trust account deposits directly. That is a separate service from what InmateAid offers. InmateAid's services cover phone savings, mail, letters, photos, postcards, greeting cards, and magazine subscriptions. For putting money on your inmate's books you will need to use a dedicated deposit service.
On JPay versus MoneyGram, both are legitimate and widely used but they work differently and are not available at every facility.
JPay is a platform specifically built for the corrections industry. It handles deposits, messaging, music, and video for a large number of state prison systems. If your inmate is in a state that contracts with JPay, it is often the most seamless option because everything runs through one account. The downside is that JPay is not available everywhere and their fee structure varies by state and deposit method.
MoneyGram is InmateAid's recommended partner for deposits and has been in the money transfer business longer than almost anyone. They service federal, state, and county facilities across the country, which gives them broader coverage than JPay in many cases. They are reliable, widely trusted, and accessible through thousands of retail locations as well as online.
The practical answer is to confirm which services your specific facility accepts before setting up an account with either one. Call the facility or check their website to see which deposit platforms they work with. In many cases both are accepted and you can choose based on fees and convenience. If only one is available, that decision is made for you.