Dig in lots of sharp sand, horticultural grit and organic material to improve the drainage.

Spritz's advice is good but is this a permanently boggy area? In which case even with the addition on sharp sand, grit and organic material you could still have a problem growing some veg. Carrots and parsnips for example need a fine, preferably sandy, tilth. Legumes should do reasonably okay but only if you add the sand etc. You may need to look at drainage.

Avoid treading on the parts that you grow veg on, make pathways so you can reach the centre of each bed without treading on the soil and compacting it.