I have read that LPA requires continuous (sometimes ordinal) and LCA uses categorical, but wasn't sure if that criteria was for DVs or IVs.
I sense some confusion here. First, LCA/LPA is applied to a set of items (or questions, or indicators). The usual notation I've seen calls these things $Y_i$, where $i$ indexes which indicator. In some sense, the indicators are the dependent variables for the LCA/LPA. That may or may not mean they're your dependent variables or your independent variables. Your post could be read as if you're thinking of applying LCA/LPA to either your vignette responses or your independent variable.
Now, suppose that my independent/predictor variable is composed of a few measures that are ordinal (typical scales using 1 "Strongly Agree" to 7 "Strongly Disagree").
Usually when I hear "scale", I think of a number of separate questions with Likert-type response scales (e.g. 1 to 7). Most people will sum the responses in a scale and treat it as continuous. If you have several separate questions and you're treating each as categorical, that's another approach.
Now, to answer your original question,
If I want to use some form of latent analysis to group people based on their response style, would it be more recommended that I use Latent Profile Analysis (LPA) or Latent Class Analysis (LCA)?
It's technically LCA. I've usually heard LCA applied to binary or categorical items, and LPA applied to continuous items (which are usually treated as Gaussian). However, conceptually, they're doing the same thing. You can even conduct an LCA/LPA on mixtures of item types (usually called mixed mode LCA, I think). The items don't even have to be Gaussian - R's
flexmix and Stata 15's
gsem are two packages/commands that I know will allow you to mix in things like mixtures of Poisson or negative binomial distributions.