Say we have a poisson process of people arrivals with rate $X$. By time $T$, we wish we have seen at least $100$ people arrive.
I'm thinking this is $1 - \Gamma(XT, 100)$. Is there any simplification possible? It seems to be just a sum of a Poisson random variable with distribution $XT$ up to $n=99$ for the people.
I don't think you're correctly using $\Gamma$ here. Let's use $\lambda$ as the rate of the process instead of $X$ for aesthetic reasons. Note $$P(N(T) \geq 100) = P(N(T) - N(0) \geq 100) =P(\operatorname{Pois}(\lambda(T-0))\geq 100) \\= P(\operatorname{Pois}(\lambda T) \geq 100) = \cdots$$