It appears from the example that the 'Network name' is a header, and are multiple 'Interface records', but it's not clear whether there could be multiple Network Name/Interface groups per file? The following assumes not.

It would also appear that could be more than one "address above Monitored circuit.", going by the "Number of addresses" field. I don't think either of the other solutions would cater for that? If so, this might be useful. I've also captured and output the Number of IPs value, but that is easily removed or discarded.

You'll note that this makes no attempt to verify the format or legality of the ip addresses whilst parsing the file, simply treating them as whitespace delimited tokens. If you need to verify the addresses, that is better done as a separate step.

#! perl -slw
use strict;

my( $nName, $iName, $vrid, $state, $paddr, $nAddr, $oAddrs, @oAddrs );

die "File does not contain a Network name header"
    unless ( $nName ) = <DATA> =~ m[Network\sname \s+ (\S+)]x;

local $/ = "\nI";
while( <DATA> ) {
    if(
        ( $iName, $vrid, $state, $paddr, $nAddr, $oAddrs ) = m[
            (?=.*? nterface           \s+ (\S+) )
            (?=.*? VRID               \s+ (\S+) )
            (?=.*? State:             \s+ (\S+) )
            (?=.*? Primary\saddress:  \s+ (\S+) )
            (?=.*?
                Number\saddresses:    \s+ (\S+)
                ( .+? )
                (?=Monitored\scircuits:)
            )
        ]smx
    ) {
        @oAddrs = split ' ', $oAddrs;
        print join ', ', $nName, $iName, $vrid, $state, $paddr, $nAddr
+, @oAddrs;
    }
    else {
        warn "File did not match required format";
    }
}
[download]

Output

C:\test>junk1
A, WEB_A, 145, Master, 194.551.205.250, 1, 135.51.045.242
A, WEB_B, 145, backup, 19.51.205.20, 2, 135.51.05.22, 321.654.987.111
A, WEB_C, 145, backup, 19.51.205.20, 3, 135.51.05.22, 123.456.789.001,
+ 321.654.987.111
[download]

Extended dataset

__DATA__
Network name A
Interface WEB_A
    ggggggggg
        VRID 145
        State:        Master
                   XXXXXXXXXXXX
    Effective  Priority:    200
        07:09:56:04:0t:ef
        Primary address:    194.551.205.250
        Number addresses:    1
        135.51.045.242
Monitored circuits:
            vbgghdfgdfg
Interface WEB_B
    ggggggggg
    Agggggg
    VRID 145
        State:        backup
                   XXXXXXXXXXXX
    Effective  Priority:    200
        07:09:56:04:0t:ef
        Primary address:    19.51.205.20
        Next advertisement:    0
        Number addresses:    2
            135.51.05.22
            321.654.987.111
        Monitored circuits:
            vbgghdfgdfg
Interface WEB_C
    ggggggggg
    Agggggg
    VRID 145
        State:        backup
                   XXXXXXXXXXXX
    Effective  Priority:    200
        07:09:56:04:0t:ef
        Primary address:    19.51.205.20
        Next advertisement:    0
        Number addresses:    3
            135.51.05.22
            123.456.789.001
            321.654.987.111
        Monitored circuits:
            vbgghdfgdfg
[download]