On the local computer
Kind of.
Running a program directly from an HTTP URL is not possible in Windows... however, it is possible to run programs from a WebDAV server (which is still HTTP with some extensions) because Windows has a WebDAV "redirector" alongside the more commonly used SMB.
For example, SysInternals offers this service at live.sysinternals.com which runs WebDAV. Although you cannot simply run https://live.sysinternals.com/disk2vhd.exe on Windows, you can rewrite the URL into an UNC path \\live.sysinternals.com\disk2vhd.exe – Windows will automatically detect that it doesn't support SMB but does support WebDAV, and will fetch & run the executable.
Otherwise, just download it to a temporary folder and run manually. (PowerShell has commands for downloading files.)
On the web server
No.
If you are interacting with a web server, purely through HTTP, then you are limited to what the web server allows you to do. You only make a GET or POST request for an URL – the webserver decides what to do.
In a sense, many websites are programs (for example the Super User site is a giant program written in something.NET); but they run only because the web server is configured to allow them to run.
For example, in the past it used to be very common to allow running any type of program if it was placed in the cgi-bin directory (regardless of its type or extension). Nowadays many webservers are instead configured to run any *.php file as a program (regardless of location).
But this usually does not extend to miscellaneous .exe files in random directories. If your requests for the URL simply download the file, then that's the only thing you can do: you, an outsider, cannot override the webserver's configuration.
(And of course, in order to run Windows .exe programs, the server has to be running Windows...)