It can be a problem as we found out early on this year. The only month that we didn't go was December due to the Decembery type things that happen in that month. January and February can be a bit troublesome finding places to stay, we have 5 & 7 year old girls and it can be difficult entertaining them in poor weather. Earlier this year we booked into a couple of CC sites on and around the south coast but although some had nice play areas that doesn't really help when it is pouring down. As GabsGD pointed out many of the sites that are open are quite basic. The other problem we found is that although some sites with facilities were open year round, they were in demand and therefore got fully booked quite early. I think that we will be looking to take a long weekend at some point over the February half-term and save the week away for the October when there is far more choice to be had (West Bay this year).
As Mel pointed out there are a number of site finding sites, UK campsite is quite good as they have pragmatic and helpful reviews of site. Best of luck with your search.