We say that a vector $\mathbf{X}$ in $\mathbb{R}^n$ is subGaussian with parameter $s$ if for any unit vector $\mathbf{u}$ and any $t \in \mathbb{R}$ we have $$ \mathbb{E}\left[ \exp(2 \pi t \langle \mathbf{X},\mathbf{u} \rangle \right] \leq \exp \left( \pi t^2 s^2 \right)$$
Let $\mathbf{X} \in \mathbb{R}^n$ be a vector such that each of its components is subGaussian with the same parameter $s$ (we may have dependent components).
Is the whole vector $\mathbf{X}$ subGaussian? If yes what is its parameter?
Your instinct is correct. If $X_1,\ldots, X_n$ are mutually independent, $s$-subguassian random variables, then $X=(X_1,\ldots, X_n)$ is $s$-subgaussian. $$ E[\exp(2\pi t \langle X, u\rangle)] = E[\exp(2\pi t \sum_{i=1}^n X_i u_i)]=\Pi_{i=1}^nE[\exp(2\pi tX_i u_i)] \le \Pi_{i=1}^n\exp(\pi u_i^2t^2 s^2)$$ $$=\exp(\pi t^2 s^2 \sum_{i=1}^n u_i^2) = \exp(\pi t^2 s^2)$$