At first glance, the behavior you're describing is consistent with having the account set up as a POP account rather than as an IMAP account. The POP protocol just grabs messages from your online inbox, and optionally deletes messages from your online inbox, there is no other synchronization. The IMAP protocol creates a local mirror copy of your online folders, and you can manage which folders are synchronized.

If you need to re-do your setup, please see this article for suggestions, and it wouldn't be a bad idea to make a backup: Switch from POP to IMAP account.

Thanks for the reply, my setup is already in IMAP :(

Post your troubleshooting info from the Thunderbird help menu.

Application Basics

   Name: Thunderbird
   Version: 45.1.1
   User Agent: Mozilla/5.0 (Windows NT 10.0; WOW64; rv:45.0) Gecko/20100101 Thunderbird/45.1.1
   Profile Folder: Show Folder

             (Local drive)
   Application Build ID: 20160526140950
   Enabled Plugins: about:plugins
   Build Configuration: about:buildconfig
   Memory Use: about:memory

 Mail and News Accounts
   account2:
     INCOMING: account2, , (none) Local Folders, plain, passwordCleartext

   account3:
     INCOMING: account3, , (imap) email.vanderbilt.edu:993, SSL, passwordCleartext
     OUTGOING: , smtpauth.vanderbilt.edu:587, alwaysSTARTTLS, passwordCleartext, true

 Crash Reports

 Extensions
   Lightning, 4.7.1.1, true, {e2fda1a4-762b-4020-b5ad-a41df1933103}
   MinimizeToTray revived (MinTrayR), 1.1.2.1-signed.1-signed, true, mintrayr@tn123.ath.cx
   Thunderbird Conversations, 2.11.4, true, gconversation@xulforum.org

 Important Modified Preferences

   Name: Value

     accessibility.lastLoadDate: 1466994486
     browser.cache.disk.capacity: 358400
     browser.cache.disk.filesystem_reported: 1
     browser.cache.disk.smart_size_cached_value: 358400
     browser.cache.disk.smart_size.first_run: false
     browser.cache.disk.smart_size.use_old_max: false
     dom.apps.reset-permissions: true
     extensions.lastAppVersion: 45.1.1
     font.name.monospace.el: Consolas
     font.name.monospace.x-cyrillic: Consolas
     font.name.monospace.x-unicode: Consolas
     font.name.monospace.x-western: Consolas
     font.name.sans-serif.el: Calibri
     font.name.sans-serif.x-cyrillic: Calibri
     font.name.sans-serif.x-unicode: Calibri
     font.name.sans-serif.x-western: Calibri
     font.name.serif.el: Cambria
     font.name.serif.x-cyrillic: Cambria
     font.name.serif.x-unicode: Cambria
     font.name.serif.x-western: Cambria
     font.size.fixed.el: 14
     font.size.fixed.x-cyrillic: 14
     font.size.fixed.x-unicode: 14
     font.size.fixed.x-western: 14
     font.size.variable.el: 17
     font.size.variable.x-cyrillic: 17
     font.size.variable.x-unicode: 17
     font.size.variable.x-western: 17
     mail.openMessageBehavior.version: 1
     mail.winsearch.firstRunDone: true
     mailnews.database.global.datastore.id: c3150056-de3a-4595-b16f-59d218dfdf3
     network.cookie.prefsMigrated: true
     network.predictor.cleaned-up: true
     places.database.lastMaintenance: 1466896260
     places.history.expiration.transient_current_max_pages: 104858
     plugin.importedState: true

 Graphics

     Adapter Description: Intel(R) HD Graphics 520
     Vendor ID: 0x8086
     Device ID: 0x1916
     Adapter RAM: Unknown
     Adapter Drivers: igdumdim64 igd10iumd64 igd10iumd64 igd12umd64 igdumdim32 igd10iumd32 igd10iumd32 igd12umd32
     Driver Version: 20.19.15.4300
     Driver Date: 10-1-2015
     Direct2D Enabled: false
     DirectWrite Enabled: false (10.0.10586.0)
     ClearType Parameters: Gamma: 2200 Pixel Structure: R ClearType Level: 100 Enhanced Contrast: 400
     WebGL Renderer: Google Inc. -- ANGLE (Intel(R) HD Graphics 520 Direct3D11 vs_5_0 ps_5_0) -- OpenGL ES 2.0 (ANGLE 2.1.0.316930d51ea9)
     GPU Accelerated Windows: 0

     AzureCanvasBackend: skia
     AzureSkiaAccelerated: 0
     AzureFallbackCanvasBackend: cairo
     AzureContentBackend: cairo

 JavaScript

 Incremental GC: 1

 Accessibility

   Activated: 1
   Prevent Accessibility: 0

 Library Versions

     Expected minimum version
     Version in use

     NSPR
     4.12
     4.12

     NSS
     3.21.1 Basic ECC
     3.21.1 Basic ECC

     NSS Util
     3.21.1
     3.21.1

     NSS SSL
     3.21.1 Basic ECC
     3.21.1 Basic ECC

     NSS S/MIME
     3.21.1 Basic ECC
     3.21.1 Basic ECC

Go to File-Subscribe and see if you have subscribed to the server Sent folder.

Go to Account Settings-Copies and Folders and see where you have set Thunderbird to save Sent messages. Is it the local sent folder or the one on the providers server?

Thank you Airmail it worked! I really had no idea how Thunderbird worked. Thank you for pointing out the subscribe feature. It turns out my Webmail has "Sent Items" folder but no "Sent" folder. Weird huh?