Assume this species is naturally born and has evolved to not have arms. The members of the species are extremely capable of performing tasks like opening doors with their feet and mouth. Also, this species lives on a planet extremely similar to Earth, and have built a civilization almost exactly the same as ours. However, while human children (and some adults) use arm wrestling as a playful test of strength, our unnamed species unfortunately has no hands. Keeping in mind that they can accomplish most basic tasks with their feet (and sometimes the assistance of their mouth), what type of activity could serve as a suitable replacement to arm wrestling?
There is an option that already exists...it's called Indian Leg Wrestling.
The two contestants lie on their backs beside each other, facing opposite directions, hips roughly on-position with each other's shoulders, and hook their right or left legs together at the knee.
The goal is to flip your opponent into a backwards somersault.
It looks like this:
A bit of googling will give you videos, tactics, and other info.
Since leg wrestling has already been covered I will have to move down a step
Possibly a not entirely serious sport, but a sport none the less.
Toe wrestling is similar to arm wrestling. To play, players must take off their shoes and socks as the game is played with bare feet. It is common courtesy for each player to remove the other player's shoes and socks. Players must link toes and each player's foot must touch flat on the other person's foot. The opponents proceed to attempt to pin (capture or trap) the other's foot for three seconds while avoiding the same. There are 3 rounds played on a best 2 out of 3 basis. First with the right foot, then left, and right again if necessary.
Just look in the animal kingdom to see all sorts of examples, mostly in mammal mating displays.
Some species use loud voices.
Some use ornate dances.
Some decorate homes.
Some smash into each other head first.
There are all sorts of similar displays of skill or strength that are possible without arms. For example imagine some sort of tug of war between two people each holding the rope in their teeth. Or a form of wrestling where they try to barge each other out of a ring. The possibilities are endless...