The foot is very complicated and there can be some very complex problems even starting way further up the body. So everything I say is very general.
I take it that this is the only time your arches get sore. Nevertheless you may want to see a podiatrist to check your feet and the way you walk. That is my biggest recommendation. If you are overpronated or walk funny that could affect the muscles and tendons in your legs and feet and thus lead to this during your workout.
You can do this flexibility test
for the ankles.
Could be your calves are excessively short/tight and it's the simple pulling on the arches. That doesn't mean that they are not tight because of another underlying problem but stil it gives your something to go on. Keep in mind that the test is talking about ankle flexiblity but imo that is a misnomer. The ability of a joint to move through a proper range of motion is better termed "mobility" since joints are not really meant to 'stretch' or bend as the term flexiblity implies. So what this test may show you is whether you calves are tight which would impair ankly movement or whether you ankles are not moving. You should be able to tell the difference.
In any case some stetching of the calves after you work them may help. You also may want to work the front of your lower legs which can also help. Nothing heavy. You can just take a light dumbell between your feet and do a sort of ankle curl with it. Do not overdo it because it is easy to get some tendonitis from overworking it.
Another thing is to get a rubber ball like a lacrosse ball or the like and use that to work and massage the bottom of your feet. When you find tender or painful spots you can focus on them. This can help a bunch.