Is it possible to change where Wireshark obtains its updates from?

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If you're referring to the automatic update feature available with Wireshark on the Windows platform, then yes, it's possible. It will require that you:

• Modify and recompile Wireshark yourself, including creating a new installer (or installers if you want to provide updates for both 32-bit and 64-bit platforms). The code you must modify is the update_url_str in the software_update.c source code.
• You will need to have a web server running on the hosting machine.
• You will have to provide a replacement stable.xml file on your server.

You can see examples of Wireshark's stable.xml files at:

• 64-bit: https://www.wireshark.org/update/0/Wi...
• 32-bit: https://www.wireshark.org/update/0/Wi...

Example page source (for 64-bit stable.xml):

<?xml version="1.0" encoding="utf-8"?>
<rss version="2.0" xmlns:sparkle="http://www.andymatuschak.org/xml-namespaces/sparkle">
<channel>
    <title>Wireshark Stable Release</title>
    <link>https://www.wireshark.org/download.html</link>
    <description>The latest stable release of Wireshark.</description>
    <language>en</language>
    <item>
      <title>Version 2.6.4</title>
      <sparkle:releaseNotesLink>https://www.wireshark.org/update/relnotes/wireshark-2.6.4.html</sparkle:releaseNotesLink>
      <pubDate>Thu, 11 Oct 2018 22:40:34 GMT</pubDate>
      <enclosure url="https://1.na.dl.wireshark.org/win64/Wireshark-win64-2.6.4.exe"
                 sparkle:version="2.6.4"
                 type="application/octet-stream">
      </enclosure>
    </item>
  </channel>
</rss>

Refer to the WinSparkle web site for more information, in particular the section on Appcast Feeds.