This is my industry. I handle European marketing (with links into sales & product management) for a huge international car parts manufacturer.
I am not sure if you want to sell online why you are asking where factors buy from, it's not necessarily going to help. But the answer is it varies, based on the factor size but also the product. You will generally find it split as follows for key product groups - braking, filtration, batteries, steering & suspension etc.
- HUGE Chains - e.g. Euro Car Parts. These are part of a multi-billion $ group (LKQ). They have their own UK distribution centres and have direct supply deals with all of their suppliers either direct from factory or via the suppliers UK or EU warehouses. The buying is being done more and more on a European level rather than by UK buyers. They will also usually have their own brands (cleverly branded up to look like some German or Italian brand they they are "exclusive" on when in fact they have branded it themselves out of China).
- Larger Independent Chains - Perhaps 20 or 30 branches, an example could be Arnold Clark Autoparts. Or MPD as mentioned above, until very recently they were the largest independent in the country but have been bought by a South African company. These will generally also deal directly with manufacturers but will probably have some UK based national distributors supplying them with infill stock as well as lower value/low running product groups. FPS are delivering multiple times a day into factors, offering same day delivery and are the biggest national distributor by far. Others are available mainly offering next day delivery. These usually also won't have a separate distribution centre like ECP do although they may designate 1 or 2 or 3 branches as "hubs" where they receive most of their deliveries and then farm it out to their other branches from there.
Then with the smaller independents (generally single branch or a couple of branches) they will be independently owned but will be part of a national buying group (A1, IFA, GAU, CAAR, etc). Apologies if I'm teaching granny to suck eggs but a buying group is basically where all these independents club together and get the suppliers to tender for their combined business. So in theory they have a direct account with the supplier but technically it is the group, rather than the factor, that has the account. So lets say a member in A1 is buying NGK spark plugs directly from NGK. If they leave the group and to go IFA, if IFA don't have an agreement with NGK then they won't be able to continue buying.
The situation then gets even more complicated because there are two types of buying group - some have their own warehouse, in which case the suppliers deliver in bulk only to the warehouse and then the warehouse splits it out to the individual members. In other cases there is no warehouse and the supplier has to send it directly to each branch which is not always possible if a supplier is only geared up for bulk pallet orders, not single box/single item picking.
These smaller independents won't necessarily have agreements with suppliers for ALL products. They will rely much more heavily on the UK national distributors such as FPS mentioned above and others. These can offer more rapid service, lower MOV's, and a much bigger basket of goods. Rather than dealing direct with a battery manufacturer, a wipers manufacturer, a spark plug manufacturer etc. you could place a £200 order with a distributor for one wiper, one battery, one spark plug, one set of wheel trims, half a dozen air fresheners and get it all delivered next day if not same day.
Even the larger chains up to the size of ECP will use the national distributors in SOME cases. Usually for very distress purchases or for consumables, generally very low value items with high SKU count. Think for example things like key fob batteries (each blister packed in a single pack), hose clips, fuel filters, fuses, battery terminals etc.... it's just not a key product line worth investing time and stock in and it's easier just to buy from a distributor who focuses more on those products.
As somebody said above, historically there was a clear hierarchy:
MANUFACTURER > NATIONAL DISTRIBUTOR / IMPORTER > LOCAL DISTRIBUTOR > FACTOR > GARAGE > CONSUMER
This has just been eroded away really and become more and more complex. People have completely skipped the chain in both directions - e.g. ECP are a factor, but they will buy directly from a manufacturer so potentially cut out the two steps in before them, and then also sell directly to a consumer so cutting out the step after them. It's somewhat chaotic! And that's before we even mention e-commerce. Which brings me round to my first point....
I'm not really sure how any of the above helps you wanting to sell online, but maybe it will. There are already some people doing excellent business online whether it's on their own sites or Amazon/eBay. Some of the sites that appear to be independent are far from it, behind the scenes they will have been set up by, or bought by, a much larger group.
Believe me when I say you are not the first to come along and say or think that you can "do things differently" to make an impact. I'll be honest - you probably can't. Most people start either dropshipping (and you have to have a VERY convincing case to get a supplier involved with doing that for you) or buying from the National distributors at least to begin with then trying to build to a level that they can buy direct from the manufacturer. Trust me, if you go knocking on the door of a manufacturer expecting to open an account without your ducks in a row about exactly how, what, and when you're going to do things, they will brush you off and point you in the direction of an existing distributor. And even they may take some convincing. To give you an example, we sell a low value product (typically £3-£5 invoice price), we sell in full carton quantities only (5 or 10 pieces per SKU), the full programme requires you to stock 200 SKU's, So your initial stock order might be 5 grand, and your MOV subsequently will be £2000. And we wouldn't entertain you unless you could show easily that you'd be spending an absolute minimum of around £50k a year.
I haven't even started on the issue of cataloguing and part identification yet, that in itself can be make or break for you.
I'm more than happy to answer further questions. I could put you in contact with many potential suppliers - ourselves included - covering most product groups. But without wishing to sound arrogant (In fact I think I'm about to sound like Clinton!

) that wouldn't be without understanding your position, how serious you are and what you think you can achieve. And it would be me passing your details to them rather than the other way round.
Hope this helps.