Does my 26 months old toddler develop immunity when he gets infections in his day care even though he's given antibiotics every time?

Maybe,maybe not. You build better immunity when the germ is a virus not affected by antibiotics than the random bacteria. Some bacteria can trigger an immune response, but it is often weaker, whether you give antibiotics or not.Many bacterial infections are secondary to a previous viral illness, so the immunity the kid builds over time can reduce the frequency of other infections.