Renly didn't move all that slowly. He was at Bitterbridge, halfway between King's Landing and Highgarden, 1 month and a few days after he marched from Highgarden. This was a normal pace according to the most accurate asoiaf timeline available. If he travelled at a fast pace he would have reached King's Landing at just about 2 weeks earlier, and in about 40 days if he ditched his supply train. It wasn't that he was staying at one place for weeks at a time.
This wasn't without reason, either. Tyrion observes that Renly is letting the two factions, Lannister and Stark, weaken each other. That way, he conserves his army and also gains more supporters with his progress by hosting tourneys and feasts to court allies. There's also another element. Renly is specifically said to feast at lords' holdfasts on his progress. His pace also allows his supply lines to keep up with his army. This way, he prevents his soldiers from plundering the countryside, and thus maintains good PR. In the War of the Roses, there was a monarch who was barred entry from a city because their army had been ravaging the crops and homes of peasants and the people feared they would do the same to the city. Renly is doing the opposite, he maintains a good image by not plundering and hosting feasts and tourneys instead, and has plausible deniability from starving the city (as he was so removed from King's Landing when the riots took place, the peasants blamed the Lannisters), and when he arrives at the gates he would be the one supplying food to the city, instead of demanding it, and thus would be welcomed instead of denied.
The Reach is huge, so that would be why it was able to supply his army and also, as mentioned, Renly was feasting his army at lord's homes as he travelled. The Tyrells were easily able to feed the starving city when they arrived. In fact they are so opulent Joffrey's feast had 66 courses. Also, in the time between Renly's death and the alliance between the Lannisters and Tyrells, the army was idle for months. But there was no disease or famine indicated. So I don't think it was ever a problem.