Assume I'm running Google Chrome with dozens of open browser tabs.
How can I search the text of the tab titles? As an example, assume I want to find the string: "vul"
Assuming you want to search all open Chrome tabs...
For simple searching in all the Headlines of the tabs, chrome has a inbuild functionality, Ctrl+Shift+A. It is doing exacly what you asked for.
However, for my purposes it was not sufficient, because if you have multiple tabs of different parts of the same website open, the search results will be the same.
I wrote a Plugin for Chrome to better fit this task. It has a shortcut (Ctrl+Shift+F) for opening and it searches full-text in all tabs. Try it yourself if you want.
https://chrome.google.com/webstore/detail/supersearch/ijajhcdgklelnpmjodhfhjooiljfhdpl?hl=en-US
For sake of completeness (i.e. addressing similar needs on other browsers), here's the link for info on doing the same on Firefox: https://support.mozilla.org/en-US/kb/search-open-tabs-firefox
NB: I just joined the community -- rules recommend entering answers first! (before I can enter a comment on another answer, or even UPvote another person's answer). I wanted to UPvote Mike Pennington's answer (or provide a complement on it in a comment); that answer is something I needed to know 20 years ago!