Moving server to new state/ISP, easy?

Latest version on Synology DS220+ NAS.

Having to move to another state and a different ISP. Anything special I need to do in order to ensure my “customers” can just launch the app on their TVs and have it find my server after it is running again?

Your new ISP does need to assign you a publicly accessible IPv4 address.
Many (if not most) don’t do that anymore, particularly on new contracts.
Ask your prospective new ISP whether such a thing is available.
Many require you to buy a “fixed public IP” additionally to get that.