The usual practice with wet ground would be fell in the winter and extract when the ground is firmer in the summer. I have heavy clay which is soft and slippy in the winter but like rock in the summer, so that's what I do with my big oaks. Excavators are more stable on slopes and wet ground as they spread the load more. Its kinda how long is a piece of string question as it'll depend just how soft and deep the wet stuff is. If its really boggy it might be a job for pulling it out with cables from a firm base at a distance. You've nothing to lose from asking a contractor to have a look, find how they'd do it, and ask for a quote. It should then be clearer what the best method is and whether you want to tackle it yourself or get in a professional. There's a big international market for stolen plant as some of it is worth tens of thousands, and lots of it 'disappears', even just bits taken off like the hydraulics and batteries, so its not something you want to leave standing around without adequate security. £800 Million worth of plant is nicked every year in the UK, so ccTV and secure premises is pretty much essential.