You can remove the newlines with chomp, then join to make a single line:

foreach $member (@memberlist) {
    open(FILE, "$member.dat") or die $!;
    lock(FILE);
    @members = <FILE>;
    chomp(@members);
    push @allmembers, join( " ", @members);
    unlock(FILE);
    close(FILE) or die $!;
# remove this
#      foreach $line (@members) {
#          push(@allmembers, $line);
#      }
}
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Update: Fixed a blunder, props to wog.

Update2: Should have also corrected your open and close, props to tilly.

After Compline,
Zaxo