Hey Mike,
The following info is pulled from an article by the SBA. I am only supplying the info as an aid to help you get closer to your answer. This is not an actual answer, and I did not sleep in a Holiday Inn Express last night. Lol.
I'll give you a link to the page, and I am going to bold the part that seems to apply to your question. This article is to help a business person decide if they want to go into a franchise. One of the questions relates to your own question.
Reading below, it seems whether or not the Pizza place must buy from a
particular supplier, or are able to contract their
own suppliers will vary according to their particular contract with the franchiser.
The answer, it seems, will be in the contract between the Pizza place and the franchiser.
You'll likely need to decide if you are going to establish a company policy on how you will handle these situations, such as: A photocopy of that portion of the contract for the job/customer file (hey, there may be future orders if it is allowable, and one copy covers all future orders as well.) Will you decide if they say it is OK in the contract, that is good enough for you. Will you have a lawyer review it? Whatever you decide is up to you. Good luck to you, here is the link:
Evaluating Franchise Opportunities: Management and Planning Series
PS:
If your shirts are excellent, and the franchise down the road decides to get jealous and try to stop you and claim you aren't allowed to make those, would it be nice to have a copy in writing showing you are? I notice in the article below, one of the questions is competing with other franchises. Again, good luck to you, I hope it works out no matter what route you take. 
What Is the Franchise Package?
Bring all your information and resources together as you examine the contract. Think carefully about the level of independence you will maintain as a franchisee. How comprehensive are the operating controls? Be very clear about the full costs of purchasing the franchise. Involve your attorney, accountant and/or business advisor as you examine these questions.
- What is the full initial cost? What does it cover?<LI style="LIST-STYLE-TYPE: none">
- Licensing fee?
- Land purchase or lease?
- Building construction or renovation?
- Equipment?
- Training?
- Starting inventory?
- Promotional fees?
- Use of operations manuals?
- What ongoing costs are paid to the franchiser?<LI style="LIST-STYLE-TYPE: none">
- Royalties?
- Ongoing training?
- Cooperative advertising fees?
- Insurance?
- Interest or financing?
- Are you required to purchase supplies from the franchiser or a designated supplier? Are the prices competitive with other suppliers?
- What, if any, restrictions apply to competition with other franchises?
- What are the terms covering renewal rights? Reselling the franchise?
