Is it appropriate to do a study using only one trait from a larger personality inventory (e.g., Extraversion)? I'm looking to gather peer reports and self reports of extraversion. Can I send out an item inventory that only has extraversion related items?
Yes. It is quite common to do a study that measures just one personality trait. There are many variations in this and reasons for doing this. Here are some examples:


You measure the Big 5, but you want to incorporate some other personality trait not captured as well by the big 5 (e.g., honesty/humility from HEXACO; dark triad, etc.)
Personality is a minor component of the study, so you just measure the one or two dimensions of interest.
You are specifically interested in the facets of a particular dimension so for example, you measure conscientiousness as well as many different facet scales of conscientiousness.


In these cases, items are often taken from a larger inventory. E.g., you take just the extraversion items from a measure of Big 5.

There are legitimate reasons for why you might want to only measure one personality trait. That said, if length is not a major issue, I would strongly encourage you to measure a broader set of traits. In particular, there are many benefits to getting a complete measure of the Big 5 (or perhaps HEXACO). You can then examine correlations in context and provide a much more complete picture.