Yes, you do need a solicitor when remortgaging your house. Depending on which lender you are seeking finance through will depend on the conveyancing company you can use. Property conveyancers need to be on the panel of recommended solicitors for the mortgage firm you are using. If they are not, they are able to apply to the lender to see if they will accept them to carry out the remortgage conveyancing. This can take time though so you are best off finding out if they are before you hire one.
I re-mortgaged my home late last year and I was required to hire a property solicitor. The mortgage broker I used recommended one to me that was on the lenders panel. They were quite expensive though, so you would be better off comparing re-mortgage property solicitors on this site to get a better deal.