Hello! Welcome. Thank you for sharing your situation.
I don't see any reason why not having been sexually active should particularly be an impediment to conceiving. (Otf it helps at all, know that I am very single. I have had male partners in the past, but not for a looooong time. I know that's not quite the same, but still.) If you do home insemination, the usual method is to use either a needleless syringe or a soft cup to get the sperm close to your cervix. For comparison purposes, if you use tampons, the syringe option is smaller than a tampon. (And sperm, obviously, are teeny and will then swim pretty much wherever).
I would sound a note of caution about home insemination, which is that it can end up expensive and doesn't necessarily give you the best odds of success. I have done it myself in the past, so I don't have any issues with it in theory (and I did, eventually, become pregnant in this way, although unfortunately I miscarried) but my experience was that it got to be very expensive (probably £1k+ per cycle to be in with a decent chance) and the timing can be stressful, particularly when you can't tell for sure exactly when you're ovulating, and have to try to interpret signs that you are about to and signs that you recently have, and hope you're interpreting them correctly. So, on the one hand I really liked the relatively low-key option of doing it at home, but on the other I spent a lot of money over my cumulative Cryos attempts, and on some cycles it was wasted because I fluffed the timing (although generally I was pretty accurate).
So, if you do want to go ahead with home insemination, it's going to be really important to monitor all the signs you can (temping, ovulation prediction kits, cervical mucus, etc), for a good few months before you first try, to get the best sense you can of how to time things.
I would also always recommend having an initial work-up at a fertility clinic to check out a couple of things before going ahead with home insemination (especially when you're paying for the sperm) - specifically, that you're ovulating (which they can check with a combination of ultrasound/blood test), and that your fallopian tubes are clear so the egg can get through (via a test involving scanning you while introducing a contrast medium into your uterus through your cervix). If you're going to be spending large amounts of money on donor sperm, I think it would suck to later find out that it couldn't have worked because there was a problem with one of those things. It would cost a few hundred pounds to have these tests, and I'd suggest waiting and doing them after you've achieved your weight loss, as this would give the best picture of your situation.
Also, be aware that Brexit may throw a spanner in the works here, because at present the legality of having sperm delivered from a sperm bank for home insemination is that the EU allows free movement of goods so although the HFEA don't really like the idea, they can't stop you. In a couple of years once we get around to Brexitting, that's likely to change and it's unclear whether it will still be possible, or whether customs will become involved.
With all of that in mind, I'd also suggest thinking about pursuing treatment through a clinic. This would probably involve slightly more weight loss than you're currently planning on (at my clinic, they treat women with a BMI up to 35) but in terms of the costs, it would probably not be all that much more expensive than home insemination, and it's likely it would improve your odds. (It's hard to know for sure, because there's no real easy of knowing what the odds of home insemination are, but a clinic has access to a wider range of ways of knowing when you're ovulation, which is the key thing. At best, the odds of home insemination working would maybe be similar to a clinic IUI, which for your age range is about 15%, but I've always felt the home odds must be a bit lower). It's obviously totally up to you, but maybe something to consider.
Well done on your weight loss so far! And best of luck to you.