No. VAT registration is not a requirement for asset finance, and plenty of businesses that are not registered are funded every day. It can be relevant to how the VAT on the purchase is treated and to the paperwork a lender asks for, but it is not in itself a condition of being considered.
Where it does come into play is the cash flow. On a hire purchase agreement the VAT on the asset is usually payable up front, and a VAT-registered business can normally reclaim it on its next return. If you are not registered, that VAT is a real cost rather than a timing difference, so it needs to be built into your budget from the start.
VAT registration can also be a useful piece of evidence for a lender, because VAT returns give a picture of turnover. If you are not registered, other proof of trading — bank statements or accounts — does the same job.
Tax treatment depends on your circumstances and we are not tax advisers, so speak to your accountant about how a particular agreement would work for you.