I'm not a true Yeller belly as I wasn't born in Lincolnshire but am married to one and have lived here from 1975-1988 and again since Spetmber this year. Here are some explanations:
The Royal North Lincolnshire Militia
The building on Burton Road, Lincoln that is now home to the Museum of Lincolnshire Life was once the barracks of the Royal North Lincolnshire Militia. The officers of this regiment would wear bright yellow waistcoats on the battlefield. This made it easier for their men to spot them (let's overlook the fact that it also made them rather obvious targets...) and also earned them the name Lincolnshire Yellowbellies.
Newts and frogs
There is, apparently, a breed of newt common to the fens that has a bright yellow underside. So now you know.
Should you not be very partial to newts Marion Christy, a yellowbelly herself, was always told that it was frogs that had the yellow bellies and were rather partial to the damp conditions of the undrained fens.
Farmers
During summer the farmers would often work without their shirts on (the saucy devils). As they tended the fields they would be bent over, and get a lovely suntan on their back. Their fronts however would be in the shadows the whole time and so would stay white. The reflection of the corn is said to have given a yellow hue to their bellies.
Mail Coach
The mail coach that ran from Lincoln to London had a yellow undercarriage. Upon it's arrival in London it is said that the locals would call out "Here comes the Lincolnshire yellowbelly".
Sheep
The traditional breed of sheep in the county is the Lincoln Longwool. As the name suggests, it's fleece was, well, long. It looked rather like the sheep in question had a bad perm. These sheep would often graze in the fields of mustard that were once a common sight around Lincolnshire. As their shaggy coat dragged along the ground it would pick up pollen from the mustard flowers and give them, you've guessed it, a yellow belly.