Fairly obviously, you need to get the drivers to upload relevant documentation to your service before they can become fully-registered with it. (i.e. you need a TWO-STAGE registration process, where the first stage is get their names into your system, with the second stage being where they provide you with relevant documentation).
If you're in the UK you can require them to scan their driving licence and also to get them to share the information about any points on their licence via this link:
https://www.gov.uk/view-driving-licence
Again, if you're in the UK, you can't find out about any 'spent' convictions that a driver might have. (Only people working with children or vulnerable adults, or within the criminal justice system, etc can be required to reveal all of their criminal convictions. Everyone else has the right to have 'spent' convictions ignored). However you can require them to undergo a DBS check at 'basic' level (which only shows unspent convictions) and then to upload the resulting document to your business.
https://www.gov.uk/request-copy-criminal-record
You'll need to take great care with the way that you word your contract with drivers. It seems clear from your post that you intend them to be self-employed (or working for their own companies, as appropriate), rather than employees of your business. That's exactly what Uber intended when they offered their services to UK private hire taxi drivers though and that didn't prevent the courts ruling that the drivers are employees of Uber and thus entitled to sick pay, holiday pay, etc.
Proceed with caution!