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

Chair exercises are the best option in this case. But it’s crucial to pick the right chair. You should be able to sit with your knees bent at 90 degrees. If you use a wheelchair, make sure the brakes are applied, or the chair is fixed to prevent moving mid-exercise.
As you exercise, keep your posture upright. Remember to check your vital signs like blood pressure (in case you have high blood pressure) or your blood sugar (in case you take medication for diabetes or hypoglycemia). Exercising with weights is probably not good for you if you suffer from such issues.
Doing chair aerobics will make your heart beat faster and help you burn calories. Any form of exercise that repeats movements at a rapid pace is likely to give you the same results as a typical cardio workout.
Use a band and wrap it under your chair, couch, bed, or any other pieces of furniture suitable for performing exercises. Start slowly, and don’t expect a strong start. As your physical strength improves with every session, increase the number of activities and repeats, as well as the total workout duration.
You can find many therapy programs at different swimming pools offering exercises for people with wheelchairs. Also, if you can move your legs a little bit, you can try water aerobics exercises. At some gyms, you’ll find types of equipment customized for wheelchair users. These pieces of equipment usually function with rowing or arm bicycling. There are portable versions of these machines for in-home exercises. You may need to attach these portable machines to a table.