Printer questions (not necessarily related to printing on card) come up quite often and there seem to be a lot of fans of the Canon Pixma range. (Count me among them!).
This thread, specifically about printing on card, also seems to point towards Canon Pixma printers (especially if you disregard the recommendations for some printers, from other manufacturers, which cost well over £200):
https://www.etsy.com/teams/7718/questions/discuss/14468033/
Unsurprisingly, you're seeking cheap inks. That's likely to mean that you'll need a printer with separate cartridges for each colour. (My own Canon Pixma definitely wasn't the cheapest one available within the range when I bought it but I only pay around £20 for 4 full sets of all five compatible inks from my preferred source; they're even cheaper on eBay. Buying the multi-colour cartridges, for cheaper printers, would be much, much more expensive).
So I'd be looking at something like this:
http://www.pcworld.co.uk/gbuk/printers-ink/printers-scanners/all-in-one-printers/canon-pixma-mg5550-all-in-one-wireless-inkjet-printer-21732484-pdt.html#cat-0
which has separate inks (and which are reasonably priced):
http://inkredible.co.uk/inkredible-saver-canon-pixma-mg5550-ink-cartridges
(£30 for 4 full sets of all five inks is pretty good. I can recommend that supplier but there are cheaper offers elsewhere on the web).
A look at the full specs shows that it accepts up to 300gsm card:
http://www.canon.co.uk/Printers/Inkjet/PIXMA/PIXMA_MG5550/#p-specification