I am looking for conditions such that a process $(X_t)_t$ where the $X_t$ are $\text{iid}$ such that there is a process $(Y_t)_t$ satisfying $P(X_t=Y_t)=1$ and $t \mapsto Y_t(\omega) \text{ is continuous}.$ There is a condition given by this for general processes, but I thought that for $iid$ variables we could maybe say more and give a condition that is also necessary.
If anything is unclear, please let me know.
The answer is "essentially never".
I claim that $\{X_t\}$ has a continuous modification iff it has a degenerate (constant) distribution; i.e. iff there exists a constant $c$ such that for each $t$, $X_t = c$ almost surely.
The backward direction is obvious: take $Y_t = c$ for all $t$.
For the forward direction, suppose there is a continuous modification $\{Y_t\}$. Fix a sequence $t_n \downarrow 0$. Verify that $\{Y_{t_n}\}$ are independent (hint: the event $\bigcap_n \{X_{t_n} = Y_{t_n}\}$ has probability 1). By continuity, $Y_0 = \lim_{n \to \infty} Y_{t_n}$, which is measurable with respect to the tail $\sigma$-field $\mathcal{T} = \bigcap_{m=1}^\infty \sigma(Y_{t_n} : n \ge m)$. The Kolomogorov 0-1 law says $\mathcal{T}$ is almost trivial, and it follows that $Y_0$ is a.s. constant; there is a constant $c$ such that $Y_0 = c$ almost surely. Hence we also have $X_0 = c$ almost surely. For any $t$, since $X_t$ has the same distribution as $X_0$, we also have $X_t = c$ almost surely.
A continuous-time stochastic process in which all the random variables $X_t$ are (nontrivially) iid is a rather useless object. For one thing, it's hard to even find a reasonable probability space $\Omega$ on which to define such a process (it can't be standard Borel), and most of the events you might like to ask about turn out to be either trivial (probability 0 or 1) or non-measurable.
