Most rinks that put on shows find a place for all different levels of kids to solo, so I'm not sure you actually will need a double to get one.
What used to happen at our rink was that they would put out a group number for each of the different skating classes, and then someone with similar ability would do the solo for that group. The show would progress from the Tiny Tots and other Beginners up through the Top Ten. So the audience would see skating at all different levels, which is encouraging to help develop interest in new skaters. I've never heard of a rink that forced you to know doubles before you could perform a solo.
Now, there may be certain exhibitions where they only bring the top skaters around, but that's a different story.
To answer your question, there is no answer. There are too many variables to just say, oh it will take you X months. If you are working on loops and backspins, then those are the building blocks for good/double jumps. Some people find the stumbling block to be the axel, and it might take them even a year or more of trying. Some can do a double before the axel, though a lot of coaches will not go there in that order. Some find that it takes many months to get the double, like a 2sal or 2toe, but then the other doubles come tumbling in (2loop, 2flip, sometimes even 2lutz).
Getting coached privately will likely lesson the time that it takes you to get these under your belt.
But don't forget, there is more to skating than just landing jumps. You should be working on Moves in the Field and getting all the foundation moves/steps/patterns which will help you be a better skater all around. A jump only lasts a few seconds, and you've got whole minutes to fill in a performance. The flow and edges and connecting moves better also be top notch to present a good performance!
How long have you been skating? And do you mind my asking how old you are?