Apparently it is legal to sell a car without the V5...BUT...as you will probably realise selling privately without one usually raises huge red flags for the potential buyer. You may well have to find the original or get a replacement to complete the sale.
If the buyer is someone who knows you and trusts you, then you can sell it, but you will have to give the buyer ALL the below info - Note, f you buy a vehicle that doesn’t have a V5C vehicle registration certificate (log book), you may not be able to tax it. You’ll need to apply for a V5C logbook using a V62 application which can take 6 weeks and costs £25.
Selling your car without a V5C
When selling without a V5C, you’ll need to provide a bill of sale which includes all info for the new buyer to apply for a new logbook. That includes:
Vehicle make and model
Your car’s reg
VIN plate number
Buyer’s full name
Date of sale
Signature of buyer and seller
Price and payment terms
The fact that the vehicle is ‘sold as seen’
It may well be worth contacting the buyer and explaining the situation, but if it is a stranger, prepare yourself for him / her to walk away until you have the V5, looks like you will have to keep searching 
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