I haven't been to Frederikshavn for many years, but I'm a local in Denmark and do get relatively much around the country in both cities and small towns.
I have not used cash in Denmark for years. Many years. Literally.
A lot of locals don't carry any cash. Having a store not accepting cash would be almost (if not absolutely) unheard of, as a large part of the potential customers wouldn't be able to shop there, since we don't bring cash around.
In the very unlikely event that you will need cash, there are ATMs in Frederikshavn. There's for example a KONTANTEN ATM in "downtown" Frederikshavn near the train station.
Bring a VISA card or MasterCard. Apple Pay or similar are also widely accepted, so maybe bring that too. Then you'll be good to go.
And just for information, tipping is not expected. I understand tipping in restaurants, and can do it (occasionally/rarely) for very good service. But being from Denmark, I must say that I find the concept of tipping housekeeping staff very strange.