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Yes, it's a dependent clause—and, because it's going first, there should be a comma after it.

From "Commas with Subordinate Clauses—A Reader’s Question" at The Editor's Blog, Beth Hill says the following:

An adverbial clause often starts with a subordinating conjunction. A short list of subordinating conjunctions:

although
after
as
because
before
once
since
though
until
when
while

A subordinate clause that stands alone is a sentence fragment . . . While we can use dependent clauses as sentence fragments, most of the time we don’t. We usually pair them with at least one independent clause and create sentences . . . Independent clauses often come first in our text, but putting dependent clauses first gives us variety in sentence construction . . .

When an adverbial dependent clause comes before the independent clause, we put a comma after the dependent clause (between the clauses). We don’t have to give any consideration to the topic of essential or nonessential—when the dependent clause comes before the independent, use a comma to separate them.

Note that it's a dependent clause because it wouldn't make sense on its own unless it were in response to what somebody else said:

"As a member of the rescue team."
"Sorry, what?"

but

"In what capacity did have a terrifying experience?"
"As a member of the rescue team."

In short, in order for it to be meaningful in any way, it has to have context—either within a dialogue or linked to an independent clause in the same sentence.

In order to address some comments, I interpret as in this sentence in the same sense as because or since.

From Merriam-Webster's seventh sense of the conjunction:

7 : for the reason that : BECAUSE, SINCE
// stayed home as she had no car

I also take there to be an elided (but implied) verb.

Therefore, the sentence could be rephrased like this:

(Because / Since) [I was] a member of the rescue team, I had a terrifying experience.

It's also possible to interpret as in Merriam-Webster's fifth sense:

5 : WHILE, WHEN
// spilled the milk as she got up

Or:

(While / When) [I was] a member of the rescue team, I had a terrifying experience.

In either of these interpretations, as functions as part of a subordinate clause in a synonymous sense with one of the other words.