Is it okay to just do my $str = chomp(<STDIN>);?
No, because chomp is a bit special: it modifies its argument(s) and returns the total number of characters removed from all its arguments. The code you showed would fail because chomp wants to modify its arguments, but can't modify <STDIN> itself.
The reason chomp( my $str = <STDIN> ); works is because a scalar assignment in Perl like ( my $str = <STDIN> ) is modifiable (an "lvalue"), as described in Assignment Operators: "Modifying an assignment is equivalent to doing the assignment and then modifying the variable that was assigned to."
If you want to be a little bit more verbose, what you can do is:
my $str = <STDIN>;
chomp($str);