Have the horn be an additional method of impregnation, rather than replacing the current method
As others have pointed out, yes it would be possible, but the horn would be very exposed and, if they are also used for defence or duelling, then they are put in a lot of unnecessary danger.
What i propose is to make the horn more of an additional form of impregnation, rather than the sole method. Many animals, such as sharks and snakes, have two sets of “equipment” (though normally these are located next to each other, there is no reason they couldn’t be in different places). A unicorn would probably have similar equipment to a horse - though likely smaller if the horn is meant as the primary method of impregnation. Im not sure how feasible this is but having a secondary reproductive system as a back up seems logical - especially if the primary system is at risk of damage.
The horn might be like a snail’s love dart
Alternatively, the horn may be akin to a snail’s love dart in that it is used prior to insemination, inserting hormones into the female unicorn to create more favourable conditions to increase the chances of successful impregnation. If this were the case, I feel it would make a lot more sense for several reasons:
Firstly, although the horn is used in reproduction, much like a snail’s love dart, it is not strictly necessary - it simply increases the chance of insemination being successful. This means that having the horn exposed and using it for duelling is not as risky as they are not fighting with an organ that is necessary for their survival.
Secondly, it means that the horn can be far stronger as it does not need as much soft tissue inside of it. All it would need is a channel through the middle of it to allow fluids to pass through, much like the teeth of venomous snakes. So, if the horn is stronger, it further reduces the risk of damage and also reduces the need for it to be protected (allowing it to be very exposed on their heads).
Thirdly, because the horn is now just for inserting hormone-filled fluids, it shouldn’t need to take up as much space inside the unicorn’s head. Also, there is some real life precedent for this as the venom tubes for snakes runs through their head:
Imagine if, instead of a snake, this was a unicorn. And instead of the ducts running under the brain and through the snake’s teeth, they instead ran upwards through the inside of a unicorn’s horn.
Now proportionately, the gland would be far smaller compared to the brain of a unicorn (which would likely be similar to a horse or zebra), and it would contain hormone-filled fluid rather than venom, but it would function in a similar way nonetheless.
In regards to how insemination would actually work, the unicorn would first inject the hormones into it’s mate’s reproductive system, before continuing to mate like a horse would.
(Or, if you felt so inclined, you could have the horn be like an injection - it pierces the female’s skin to inject the hormones into her bloodstream.)