Are there any cardio exercises for seniors that mainly use the arms and do not involve walking, biking, using a treadmill, or swimming?

You can do arm circles. It’s a great way to strengthen your arms and shoulders. You can do the exercise standing up or sitting on a chair; of course, you’ll burn more calories standing up. It’s pretty simple.
Here’s the instruction: extend your arms to your side. Start making clockwise circles with your arms. You don’t have to do it quickly. Count to 30 and then reverse the circles counterclockwise. You can rest for 5 to 10 seconds and repeat the set. If you don’t have any particular problems, you can hold light dumbbells in your hands.
Pushups are also excellent for arm strength. But they’re not that easy. If you are unable to perform pushups, you can do the alternatives.
Wall pushups are accommodating. But you can’t do wall pushups sitting on a chair. You need to stand about two steps away from a wall; standing closer would make it easier.
Put your hands at shoulder height on the wall as you lean forward. Press into the wall with your elbows bent until your face is closer to the surface. After that, extend your arms fully. Repeat. I’d suggest starting with at least 15 wall pushups and developing them as you progress.