As it was only a puppet government, was there much presence of the Wehrmacht in the area of Vichy France? Especially as the opening of another front looked more and more likely?

Vichy had been occupied in 1942. So there wasn't a difference between any zones of France any more except near the German border. The French government had nominal civilian control over the country (with a lot of consultation with the Germans). The Germans had military control of the country. The southern part of France mostly aligning with what had been unoccupied Vichy. By 1944 this area was assigned to Army Group G which had 11 divisions.

If so, why didn't the push into Germany start here instead of Italy "The soft underbelly". Surely this would have been a easier target for a larger force to advance through?

As Southern France was militarily occupied it didn't hold much interest as a "easy" target. Italy made a very attractive target by comparison. After having conquered North Africa the Allies were concentrated in Tunisia. Sicily was lightly defended. The Italians were ready to come to terms and get out of the war. An invasion up Italy offered the potential to knock out Italy and tie up German forces. Both of these were realized. And most importantly Sicily was close. In 1943 the allies were desperately short on sea transport. An attack on Sicily and then Italy was achievable. An attack on Southern France wasn't.

Come 1944 it was doable. And in August operation Dragoon was launched near Marseille to add additional pressure against the Germans and to open the Med ports for Allied supplies. So in the end, the Allies did invade souther france