It would depend on how the claim was settled. With most caravan claims, the insurance company will allow you to buy from your local dealer, although they will pay the costs up to the sum insured for you. This allows you to trade up or change makes if you want to, and means there is a contract between you and the dealer.
If for any reason the insurer sources the caravan for you, then possibly the contract is with the insurer, although it is one to provide you with the caravan. It could be said that the insurer is purchasing on your behalf so the dealer still has a contract with you via the insurer, but this is a grey area and I am not aware of any cases involving this point. If the caravan is faulty that may even give you a claim against the insurer if it held that they were the purchaser, it all gets a bit complicated though.
In general the exact lost caravan will not be available as a new model except for very new one as things will have moved on, so the insurer replacing it is unlikely. Replacement by the insurer is common in the motor insurance market but far less so in caravan insurance.
The only reason an insurer would want to replace the caravan themselves would be if they thought they could get a better deal, but that would mean them getting an agreement with the trade and the numbers any one company would have to replace in a year might be too low for them to do a deal. You could imagine one of the bigger ones might get a deal from a large group like Marquis who could supply most models, whereas very few others can offer that range. The policy terms do allow them to do this if they wish though.