Of course it can cause stiff fingers. Or sore wrists, arms, shoulders, backs, hips, necks. The causes are bad posture, too much guitar weight, or over-stressing muscles and joints from overuse.
Getting the action lowered can help. If you use heavy gauge strings consider switching to some that are lighter. Some players like heavy strings for the sound, but pressing and bending strings with high action, or heavy gauge can over stress the joints and ligaments too.
Go talk to a good guitar tech, or a luthier, and see if the neck is a good fit for your hand. Everybody's hand size is different, both in the palm and fingers, which is why they make guitar necks in different radiuses and widths. Having to reach around a neck that is too wide for your hand, or across a fingerboard that is too wide for your fingers will cause RSI, leading to stiffness.
Also, though we never like to think it can happen, arthritis can set in as we age, so it might be good to talk to your doctor. They'll probably recommend ice after you play/practice to help reduce swelling, but it's better to find out for sure.
And, yes, an electric can supply lighter action/string tension, but you have to watch out for the body weight. I played a '78 Les Paul Custom for years, and it got too heavy for me, causing neck and shoulder pain. Another player in town has one also, and says it hurts his neck, shoulder and hips. So, you could jump from the frying pan into the fire if you get something too heavy.