Yes the suppliers can check them, but this is usually not done, as it can be seen like the wolf looking after the hens.  For instance if you have a maintenance contract with them that all repairs are covered by the contract, then (I'm not saying they would) but could ignore a fault which would be a financial burden to them. 

I think it is best practice to have an independant competent inspector to carry out the LOLER checks.  Hope this helps.

Yes, they can organise the inspections on your behalf. CAT or Jungheinreich will have their own partnerships with external insurance companies, which they use to carry out these checks. A LOLER schedule will be produced as a result and they will be inspected as and when required going forward. Our site operates in the same way but with a different brand of trucks.