This issue was once raised in Chrome and is documented in :
Issue 6356004: If the user selects text in the current tab, and types CTRL-F,... (Closed)
In this Issue article, six successive patches were applied to Chrome to enable this functionality.
However, because of too numerous unsolved issues, such as
line breaks and other characters that can't be printed in the find box,
the developer Simon Morris has given up and concluded :
Besides these issues, this behavior just doesn't feel terribly
compelling when testing it. Perhaps I'm too used to the existing
Chrome behavior, but while I expect prepopulation based on previous
search strings in this or other tabs, I don't really expect selecting
text to change my search string, and it doesn't feel terribly hard to
sandwich the ctrl-f in a ctrl-c/ctrl-v pair to be explicit, given that
I've just selected what I want to search for.
Since Ben already voted against this and the above testing doesn't
give me strong counterarguments, I'm going ahead and closing.
I see this as pure laziness, as this problem and others were easily solved in other browsers.
I have also searched but not found any Chrome extension that solves the problem.
The only solution I can therefore suggest, is to use a keyboard macro bound to a hotkey
to issue the series of "Ctrl+C Ctrl+F Ctrl+V".
If you are on Windows you can use a Autohotkey script for this.
See this thread for an example :
Chrome extension for highlight+find prepopulate?
An Autohotkey script can also be bound to a given process or window, so one can further fine-tune this script to apply only to Chrome.
If you are on Linux, you can use this port of Autohotkey : IronAHK, which runs on .NET or Mono for Linux/Mac.
This article might be useful : Install IronAHK on Linux.