That's not a "might."
If you are working for a company, you are working for that company.
If you are administering your SaaS tools when you're supposed to be working on a ticket, or in a meeting, this will be a problem. If you are using company resources (computer, network) while working on the SaaS, it will be a problem.
Putting it on shouldn't hurt you if you are able to assure the company that you are able to separate the two and your SaaS will not interrupt or disturb you during working hours.
Having an issue with the SaaS interrupt or disturb you during working hours could (regardless of if you put it on the resume or not) could have anything from a "that's ok", a stern talking to, a negative reference, being let go, to a talk with legal.
Personal projects (and side companies) and the company you are interning for should never overlap unless given explicit written permission by your manager - that includes time, resources, and equipment.