Is it permissible for a third party to pay the seller in cash and then collect the amount in installments from the buyer, in cases where the seller only accepts cash and the buyer does not possess the full cash price?

If the money is paid on behalf of the buyer, and then more than what was paid is taken back at a later date, this is clear usury (riba). The legitimate Islamic solution is to structure the transaction as a Murabaha sale for the one who commands the purchase, such that the owner of the money first buys the commodity for himself, and then sells it to the buyer on a deferred payment basis at the agreed-upon price, after it has come under his guarantee.