The technology used to gain access to the internet at campsites is called WI-Fi, it will be a feature on even the cheapest laptop.
Wi-Fi, "hotspots", transmitters abound everywhere however most available for public access require a subscription. In the case of those used by the Caravan Club it is extremely expensive for the casual user, its really a business man's system.
However places like McDonalds, J D Whetherspoons and many libraries provide completely free Wi-Fi. Also there are groups you can join like FON that permit you to share other individuals domestic Wi-Fi.
Back to the computer: if its only emails, a bit of web browsing and word processing that you need it for then buy the very cheapest you can find, even the sub