Usually, yes. You do not normally need to wait for the new V5C to arrive before arranging insurance, provided the ownership and vehicle details you give are correct. The important point is to make the policy describe the real arrangement rather than the simplest answer on the form.
Why this can be an insurance issue
After buying a used car there can be a period when DVLA records are being updated. Insurers are familiar with this. The key issue is that you accurately state when you became the owner or keeper and provide the correct vehicle details.
Motor insurers price a bundle of facts together. A detail that looks minor on its own can matter because it changes who controls the vehicle, where it is kept, how often it is driven or what the insurer may have to pay after a claim. That is why two insurers can react differently to the same situation.
What to put on the quotation
Keep the purchase receipt or other evidence of the transaction. If an insurer asks about the registered keeper, explain that the V5C transfer is in progress.
If the online form does not give you an answer that truthfully fits the situation, do not force the nearest convenient option. Complete as much as you can and check the point with the provider before buying. Keeping a note or email confirming an unusual arrangement can be useful later.
What can go wrong
Do not describe yourself as the registered keeper before the transfer if the question specifically asks what the current DVLA record says. Follow the wording on the quote form.
The problem with an inaccurate answer is not merely that the premium might have been different. If the undisclosed fact is relevant to a later claim, the insurer may investigate whether the policy was written on the correct basis in the first place.
Will every insurer treat it the same way?
No. Underwriting rules differ. One provider may accept the arrangement with no difficulty, another may charge more, and another may decline to quote. That variation is one reason a comparison can be useful when your circumstances are a little outside the ordinary.
Before you buy
Read the main policy information and check the excess, level of cover and any condition connected with the unusual part of your situation. A cheap quotation is only useful if the policy actually reflects the way the car is owned, kept and used.
See how different insurers respond
Enter the details as they really are and compare the quotations and policy terms offered for your circumstances.