Yes, and in Sandy that is easier than in most old towns. Very few buildings are listed. No conservation area covers the town. Most of the housing is post-war, built without any heritage constraints. The clay under the foundations will have more to say about an extension than any heritage officer.
Sandy has 36 listed buildings in total. None is Grade I, one is Grade II* and the other 35 are Grade II. Listing covers the whole building, inside and out, so changes to any of those 36 need listed building consent on top of ordinary planning permission, and a like-for-like window can turn into a discussion. Most of Sandy was built in the 1970s to rehouse Londoners, and no one has yet put a London overspill semi forward for listing.
Planning records for the area show 687 applications between April 2024 and June 2026, and 323 of them in the last twelve months. People here alter their homes all the time. With that many applications, a neighbour has quite possibly already tried something like your idea. Central Bedfordshire Council's planning register will show what was proposed nearby and what conditions came with any permission. That includes any condition on a newer estate that removed the usual right to extend without applying.
The ground is the real constraint. The British Geological Survey puts Sandy's shrink-swell hazard at class 3, and the whole area reaches that class. In practice the clay swells when wet and shrinks when dry, so an extension here may need deeper foundations than the same design on firmer ground. That makes a structural engineer's view part of the budget from the start. It is a routine matter for building control, and a builder who has worked locally will expect it.
For most houses in Sandy, the outcome is good news. Outside a conservation area, many single-storey rear extensions, loft conversions and outbuildings on a house can go ahead under permitted development rights without a full planning application, as long as they stay within the national size limits. Flats have no such rights, the 36 listed buildings need consent for almost anything, and building regulations apply to everyone. Anything bigger or closer to a boundary goes through ordinary planning permission, where the council judges the design and its effect on the neighbours.
Freedom to change your home is one of Sandy's real strengths. Unless you buy one of the listed few, the main questions about an extension are the planning department's usual ones and how deep the foundations need to go into the clay.