Hi Toffee, not mad or impossible but it maybe needs a bit of thought! I don't have any experience of being with a wheelchair user in Rome, but I live near and go quite a lot with DS age 9, so may be able to help a bit. Firstly, I am sure they would enjoy it - it is a great city for a few days, especially if they are interested in the history of it.
As regards getting around in general, I would plan for walking, with perhaps the occasional taxi if necessary. Buses are OK, usually plenty of them and not too expensive, but they are often very crowded, especially if they go through the main tourist areas, it would be very difficult to get on and off with a wheelchair. Walking should be OK, some pavements can be very narrow though, but there are lots of streets in central Rome that are virtually traffic free, running alongside the ones packed with cars. They are usually cobbled so that might be a bit difficult, but not impossible I would think. Lots of sights are also almost are all traffic free, e.g., Piazza Navona, Campo de' Fiori, Spanish Steps, area around the Coliseum, Trastevere area.
More specifically some of the sights you may want to see - which I think your DCs would really enjoy -
Coliseum - has a lift from the ground floor to the upper levels, so should be no problem at all, it is also a fairly easy walk there from the centre - not far at all.
Roman Forum - has steep slopes down into it, but I don't think any steps, and it is not too far to walk. Some parts of it may not be accessible to you but the main bits would be.
Baths of Caracalla - I think all on one level so very easy to get around, but probably too far to walk from centre - but the archeobus stops there
Mouth of Truth - just one step to go down into it, a bit of a walk up hill though back into the centre
Trajan's Markets - I am not sure about this one, I think it is accessible - I don't remember any steps but possibly some steep slopes.
There is also a great new display of Gladiators helmets etc., under the piazza Navona, in the ruins of the old stadium. I think that has access and is really interesting (if you are interested I will find out the name and you can check).
If you DS can walk up to the top deck of a bus they would probably enjoy the Archeobus, it takes you outside the city and is a really interesting trip (better on the top as it is open top). Likewise a trip around the centre on the open top bus may be a good idea.
Instead of going to Pompeii, which is huge and not easy to walk around have you thought about a trip to Ostia Antica - the old Roman Port? It is very much easier to get to, only about 25 minutes on the train from Rome, and really interesting - DS enjoyed it just as much as Pompeii and I think it is probably more accessible once you get there. For getting there though I would check it out, going on the train there are some steps, and you have to cross a road bridge but I'm sure there are other ways to get there.
Finally, I wouldn't consider going it July and August if you can avoid it - it is usually just too hot to walk around.
Sorry - this is a bit long - but hope it helps a bit