Vehicle with good traction in heavy rain and mud ??

Good traction in mud or heavy rain is provided by your properly inflated tires that have good tread on them - not by  the vehicle you're driving.

If you're driving around on a set of tires that are as smooth as a billiard ball, you won't have any traction at all - no matter what kind of vehicle you're driving. 

A pickup is fine - but if you're going to be in mud you need two things: 4WD, and good tires. Not the usual all season tires most cars/trucks come with, but I'd recommend all terrain tires. They will be much better in the mud and still be usable on the highway, whereas a pure offroad/mud tire will be terrible on pavement. 

Get a Jeep Cherokee.

Thanks for telling us, now do you have a question to ask? If you prefer a two passenger pickup truck then go get one, we do not care. 

Whatever car you already own is what I'd show up in (assuming I'd accept that job, which I wouldn't...).  I'd be surprised if it's even legal for them to require you have a vehicle with capabilities beyond what any road legal vehicle requires to get you TO work.  I mean, how many months' worth of your wages do they expect you to spend for even a cheap old 4x4 vehicle to traverse THEIR muddy worksite?  Even if that happens to be legal, it's still absurd.

If they require a 4x4 vehicle to complete your job responsibilities then I'd say it is entirely their responsibility to provide it for you on the job site.  Say the job pays (...I have no idea what that sort of job pays so please forgive me if I'm way off on the dollar amount) a nice round $4000 per month, net.  Even if you found a beater 4x4 for ~$2000 to use, that means 100% of an entire 2-week paycheck would have to go to paying for a vehicle you only needed to benefit them.  That sounds an awful lot like wage theft to me. 

I would use a Jeep Wrangler.  

Any truck with 4 Wheel drive and decent ground clearance should fit the bill