(This is how the sectioning commands of the standard LaTeX classes do it.)

Put the following in the “begin” commands of the environment:

\everypar{{\setbox0=\lastbox}\everypar{}}

At the start of the first paragraph, the \parindent glue will be removed and be put into box 0. (The braces around the \setbox command are to keep the scope of this box assignment local.) Finally we clear \everypar such that subsequent paragraphs behave normally.

Remark: One might be tempted to set \everypar{\hskip -\parindent}, but if the user has issued a \noindent command at the start of the environment body then this has an undesirable effect.