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if you look at the query string on the links from that site, you can get a pretty good idea of what they are doing.

There used to be this modification, which they probably used:
https://vborg.vbsupport.ru/showthread.php?t=102405

Since you do not have that choice, take a look at "memberlist.php" you can see that you can query "custom user profile fields", this is essentially what they are doing.
Sorry, I could not find another modification that would assist you.

If you had the specifics on what fields you wish to search, then I could be more helpful.

X vBulletin 3.8.12 by vBS Debug Information
Template Usage: (1)SHOWTHREAD_SHOWPOST (1)ad_footer_end (1)ad_footer_start (1)ad_header_end (1)ad_header_logo (1)ad_navbar_below (1)footer (1)gobutton (1)header (1)headinclude (6)option (1)post_thanks_box (1)post_thanks_button (1)post_thanks_javascript (1)post_thanks_navbar_search (1)post_thanks_postbit_info (1)postbit (1)postbit_onlinestatus (1)postbit_wrapper (1)spacer_close (1)spacer_open Phrase Groups Available: global postbit reputationlevel showthread	Included Files: ./showpost.php ./global.php ./includes/init.php ./includes/class_core.php ./includes/config.php ./includes/functions.php ./includes/class_hook.php ./includes/modsystem_functions.php ./includes/functions_bigthree.php ./includes/class_postbit.php ./includes/class_bbcode.php ./includes/functions_reputation.php ./includes/functions_post_thanks.php Hooks Called: init_startup init_startup_session_setup_start init_startup_session_setup_complete cache_permissions fetch_postinfo_query fetch_postinfo fetch_threadinfo_query fetch_threadinfo fetch_foruminfo style_fetch cache_templates global_start parse_templates global_setup_complete showpost_start bbcode_fetch_tags bbcode_create postbit_factory showpost_post postbit_display_start post_thanks_function_post_thanks_off_start post_thanks_function_post_thanks_off_end post_thanks_function_fetch_thanks_start post_thanks_function_fetch_thanks_end post_thanks_function_thanked_already_start post_thanks_function_thanked_already_end fetch_musername postbit_imicons bbcode_parse_start bbcode_parse_complete_precache bbcode_parse_complete postbit_display_complete post_thanks_function_can_thank_this_post_start showpost_complete
Messages:

X vBulletin 3.8.12 by vBS Debug Information

Page Generation 0.01171 seconds
Memory Usage 1,757KB
Queries Executed 11 (?)

More Information

Template Usage:

(1)SHOWTHREAD_SHOWPOST
(1)ad_footer_end
(1)ad_footer_start
(1)ad_header_end
(1)ad_header_logo
(1)ad_navbar_below
(1)footer
(1)gobutton
(1)header
(1)headinclude
(6)option
(1)post_thanks_box
(1)post_thanks_button
(1)post_thanks_javascript
(1)post_thanks_navbar_search
(1)post_thanks_postbit_info
(1)postbit
(1)postbit_onlinestatus
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(1)spacer_close
(1)spacer_open

Phrase Groups Available:

global
postbit
reputationlevel
showthread

Included Files:

./showpost.php
./global.php
./includes/init.php
./includes/class_core.php
./includes/config.php
./includes/functions.php
./includes/class_hook.php
./includes/modsystem_functions.php
./includes/functions_bigthree.php
./includes/class_postbit.php
./includes/class_bbcode.php
./includes/functions_reputation.php
./includes/functions_post_thanks.php

Hooks Called:

init_startup
init_startup_session_setup_start
init_startup_session_setup_complete
cache_permissions
fetch_postinfo_query
fetch_postinfo
fetch_threadinfo_query
fetch_threadinfo
fetch_foruminfo
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parse_templates
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bbcode_fetch_tags
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post_thanks_function_thanked_already_start
post_thanks_function_thanked_already_end
fetch_musername
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bbcode_parse_start
bbcode_parse_complete_precache
bbcode_parse_complete
postbit_display_complete
post_thanks_function_can_thank_this_post_start
showpost_complete

Messages:

X vBulletin 3.8.12 by vBS Debug Information
Page Generation 0.01171 seconds Memory Usage 1,757KB Queries Executed 11 (?)
More Information
Template Usage: (1)SHOWTHREAD_SHOWPOST (1)ad_footer_end (1)ad_footer_start (1)ad_header_end (1)ad_header_logo (1)ad_navbar_below (1)footer (1)gobutton (1)header (1)headinclude (6)option (1)post_thanks_box (1)post_thanks_button (1)post_thanks_javascript (1)post_thanks_navbar_search (1)post_thanks_postbit_info (1)postbit (1)postbit_onlinestatus (1)postbit_wrapper (1)spacer_close (1)spacer_open Phrase Groups Available: global postbit reputationlevel showthread	Included Files: ./showpost.php ./global.php ./includes/init.php ./includes/class_core.php ./includes/config.php ./includes/functions.php ./includes/class_hook.php ./includes/modsystem_functions.php ./includes/functions_bigthree.php ./includes/class_postbit.php ./includes/class_bbcode.php ./includes/functions_reputation.php ./includes/functions_post_thanks.php Hooks Called: init_startup init_startup_session_setup_start init_startup_session_setup_complete cache_permissions fetch_postinfo_query fetch_postinfo fetch_threadinfo_query fetch_threadinfo fetch_foruminfo style_fetch cache_templates global_start parse_templates global_setup_complete showpost_start bbcode_fetch_tags bbcode_create postbit_factory showpost_post postbit_display_start post_thanks_function_post_thanks_off_start post_thanks_function_post_thanks_off_end post_thanks_function_fetch_thanks_start post_thanks_function_fetch_thanks_end post_thanks_function_thanked_already_start post_thanks_function_thanked_already_end fetch_musername postbit_imicons bbcode_parse_start bbcode_parse_complete_precache bbcode_parse_complete postbit_display_complete post_thanks_function_can_thank_this_post_start showpost_complete
Messages: