No reason why you should not register for VAT if you are under the VAT threshold. And may reclaim the VAT for vehicles etc that you use in your business alongwith other running costs. However if you use you vehicle for personal use, you then need to charge yourself for that personal use.
And you do need to be reasonable - are really using the pickup for your business. So for example I am in finance, and couldn’t justify running a pickup for business use.
You can however have a personal vehicle, and charge business mileage at £0.45 per mile to your business. There are now many apps that link straight to Xero, quickbooks etc which make this easy. Doing a business trip - turn it on and record.
HMRC are well used to most of the tricks on VAT. You really do not want to get a VAT or tax inspection on your business as it is a costly and embuggerance filled process.
For VAT for small businesses / self employed there is the Flat Rate Scheme whereby you charge your customers VAT at 20%, but then depending on which sector you are in you pay a flat rate of say 12% of your turnover. HMRC have averaged out the costs for most small trades / sectirs.
This does make VAT returns very easy as you don’t have to account for every little expense etc.
However for your annual tax return you do - quite why they can’t do a flat rate scheme for tax as well rather beats me. It would be very simple, and save endless hours of embuggerance and cost for lots of sole traders.