Like others said, your sim can visit another lot and use their horse. You won't even need to knock on their door to get access to the horse, because it will normally be standing at the gate when you arrive.
However, you can only build your relation to the horse and your sim's riding skill. You can only participate in equestrian contests with a horse that is in your household. That said, there is nothing stopping you from moving a sim temporarily to that other household for the contests. You can enter contests only once per simday per horse though, so you would either need to stay there over several days, or allow the progress to take some time.
In my game I have a residential lot as a stable for anyone to visit an borrow a horse. Mainly for building the riding skill, though.