The autotrader website is best in my experience - go to https://www.autotrader.co.uk/ and select the more options heading and you can specify gearbox and number of seats as well as price and the distance you are prepared to travel.
Realistically at £7,000 you are going to be looking at 8 years plus old cars with 80,000 ish miles on them - this is fine as long as they have a service history and as others have said they have had cambelts changed at the recommended intervals. If there is no indication of cambelt being changed (and obviously if it has one!) then it is in the region of £400 at a local garage which you can factor into the price.
Ask about when it last serviced and check the MOT history for each car online. I You should get a car with as long an MOT as possible and ideally they will put it through an MOT test before you buy it so it has a year.
For a seven seater MPVs are probably best - these have fallen out of favour a bit in recent years as attention has moved to SUVs. But that does means they are comparatively cheaper.
One thing to consider is how often you will using the seven seats - in many 7 seaters two of the seats are quite small and maybe only suitable for children, also in many cases with all seven seats in use there is a very small boot space. For these your main choices are Renault Grand Scenic, Citroen C4 Spacetourer/Picasso or Peugoet 5008 - or Ford SMax, Ford CMax, Vauxhall Zafira or Volkswagen Touran. These are all fairly similar sized and are fairly widely available with reliable engines. As long as they have service history by this stage they should be fairly reliable and as these are major manufacturers servicing and spares will be cheap. They will also be good for fuel economy / insurance costs.
If you are going to need more space with the seven seats / the seven seats will be used by adults more you will want something a bit bigger - Ford Galaxy/ VW Sharan/ Seat Alhambra which are basically the same car.
Try and find a garage that has several suitable cars in stock - nothing beats actually sitting in to make sure everything is comfortable for you/ passengers.
One further thing to think of is the fuel type - diesel is cheaper in terms of fuel consuimption but if you are only doing around town journeys petrol is probably better as the diesel filters can get clogged (they need to go on a long journey every so often to clear the filter) plus London and many cities are banning disesels/ charging for them due to pollution issues.
We had a similar choice about 3 years ago and opted for an eight year old Peugeot 5008 with 85,000 miles on it which so far has been fine, previously we had a similar Renault Scenic. We did look at an SMax as well which felt good but was more expensive and the Touran which again was lovely but VWs are considered better made and so was more expensive. We also looked at the Volvos and Audis but these are premium brands amd were more expensive to buy and parts and insurance are more.