It depends on which side of the planning line you bought. Four conservation areas cover parts of the town. Sixty-one listed buildings sit within its bounds. Most of the housing stock is outside both. For those homes, the answer is broadly yes, within the ordinary permitted development framework.
On listed buildings, Seaford has 61 in total: 2 of Grade one, 1 of Grade two star, and 58 of ordinary Grade two. A Grade one listing puts a building among the most protected structures in the country. Grade two star is a rank down but still requires listed building consent for any material change. Grade two, the biggest category by a distance, means the same thing at a smaller scale: consent for external alterations, for interior work that affects character, for replacing windows, for taking down chimneys, for adding to the roof line. The paperwork adds real months to any project. The rewards of buying a listed building are that it looks better than the estates around it and holds its character. The costs are that you cannot fix things the way other homeowners fix them, and every conversation with a builder becomes a conversation with the local planning department first.
There are four conservation areas within the town: the Town Centre, East Blatchington, Bishopstone, and Chyngton Lane. Being in a conservation area does not stop you from extending or altering your home. It changes the process. You lose some of the permitted development rights that homeowners in unconstrained areas take for granted. Larger extensions can require full planning permission where they would not elsewhere. Roof alterations, cladding, front-elevation work, and demolition all sit under tighter scrutiny. Which side of a conservation-area boundary your house is on can materially change what you can build on the same street as your neighbour.
Outside the conservation areas and off the listed register, most Seaford homes can be extended under the standard permitted development framework: a single-storey rear extension within the size limits, a loft conversion within the volume allowance, a porch, changes that do not affect the highway. Larger schemes, side extensions, and anything eating into the space between neighbouring properties still need full planning permission. The planning department is not notably harsh by south-coast standards, and applications on non-designated houses generally get through when the design makes sense. Speed is another matter, but the answer is not the reflexive refusal that some conservation-heavy south-coast towns have become known for.
Middling freedom by south-coast standards. If you bought a Victorian villa near the seafront inside a conservation area, you already know what you signed up for. If you bought a post-war semi in the main body of the town, the planning file will treat you like an ordinary homeowner, which is the polite version of the answer.