You do need to consider the different taxes individually.
You’re not entitled to reclaim from HMRC through the VAT system any of the VAT you’re charged on goods that you buy personally, regardless of what use you then put them to.
VAT on expenditure can only be reclaimed by VAT registered businesses, and only when they have been charged it by their suppliers and the purchase is to be used by them in making taxable supplies (which essentially means generating income which itself is subject to VAT).
Employment Taxes (Income Tax / National Insurance)
This is not my current area of knowledge, I’ve not been involved with employment taxes since 2014. So you may be better with an income tax forum! I’m aware of this one https://www.taxationweb.co.uk/forum/viewforum.php?f=9 and I’m sure there must be others!
But I know that employees are entitled to reclaim income tax relief in respect of some purchases they may make for use in their employment, provided that those costs are not reimbursed as expenses by their employer. This tax relief is what is covered by the form in your original post. It’s likely that any values you’re asked for on this form should be the VAT inclusive amounts, as that is the cost that you’ve incurred.
I’m sure you can’t reclaim the actual cost of a car in this way. In virtually all circumstances HMRC consider cars to have an element of, or potential for, private use – even when they’re owned by businesses. However, HMRC do allow tax relief on 45p per mile when an employee uses their own car for what HMRC regard as business use. There are some complex rules on what counts as business use, for example journeys from home to a normal workplace are excluded. If you’re not paid a mileage allowance by your employer (or are paid at a rate of less than 45p per mile), you are probably entitled to tax relief on the 45p per mile (or the difference between that and whatever your employer does pay).