As others have said, there are childrens ponies available for share, but most people will be (rightly) a bit nervous about a novice 12/13 year old and her un-horsey Mum taking this on. This isn't meant to be harsh and your DD is obviously not a beginner in riding school terms but if she's only ever ridden school ponies under supervision she is effectively a novice with privately owned ponies who can be much more lively and challenging than school ponies who know their job well and can be swiftly corrected by adults if they start to misbehave, both when ridden and when being handled on the ground!
Essentially, at least for the first few weeks/months, you will need someone fairly experienced available to you at all times in case you run into problems bringing the pony in, grooming, tacking up, spotting injuries or tack malfunctions, dealing with any accidents etc - and since for most owners the whole point of getting a sharer is that they then don't have to be at the yard those days, it would have to come from a fellow owner, yard owner or instructor, and not all would be willing/able to do this.
Also, don't underestimate the challenge/change for your DD from riding in lessons where she will always have someone telling her what to do (now trot, now canter, now change the rein etc) to riding all on her own - children can either end up wanting to just bomb around in canter all the time (not good for the pony obviously) or just get bored/confused and stop enjoying riding as much. This can be particularly the case if there aren't other children to ride with or they are not able/allowed to hack out so are stuck in the arena. It's a good lesson for them though, teaches them how to put together a good effective schooling session with goals to work on etc., which is translatable to more than riding! But again if you are not horsey you may struggle to help her here.
I don't mean to sound really negative - you may get lucky and find a pony on a supportive full livery yard with lots of people and other children around which could work, or perhaps someone who has more than one pony so are around at the times you'd be there. These types of shares will almost certainly charge you a fair amount, compared to a share where your contribution is mainly doing yard jobs and exercising the pony. And you might need to be patient and wait for the right opportunity to come up. Or as others have said, riding school lease/share schemes are a good compromise although again, not cheap!
Good things you can be doing in the meantime are to educate yourself a bit on horse care (get a book, or better yet see if you can take a course at a local stables). You can advertise for shares in local tack shops and also on local horsey facebook groups. The best way to find a share is probably through word of mouth - can you ask your instructor if s/he has heard of anything local? Do make sure if you do enter into a private share agreement that someone experienced and independent (i.e. not the ponys owner!) has seen your DD riding the pony and agrees it is suitable/safe - your instructor may be willing to assess them together for a fee. Also make sure you get a proper written share agreement in place- the BHS have a template I think, and that your DD is covered by proper insurance - BHS gold membership comes with insurance.
While your DD is waiting, can she and her sister join a local Pony Club? There are branches for children without their own ponies, and they learn horse care as well as riding, and its good fun for them. Does your riding school (or other local ones) run pony days and weeks in school holidays? Again they usually do some stable management and more fun riding and games at these and are a good way of getting more riding in. In a few years she may be able to help out at the riding school as well which is great experience.
Plenty of pony mad children don't ever get their own pony no matter how much they want one and have to make do with lessons and rides where they can get them so please don't feel obliged to find a way for her to have her 'own' pony, they are such a huge responsibility/hassle even when sharing, no child 'needs' one. It can serve as motivation for her to get a good job as an adult so she can buy her own then
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