Proxmox suspend resume web UI buttons exist, but they’re buried under the More dropdown in the VM’s top-right corner. Hibernate isn’t supported, but suspend and resume work fine for most VMs.

Where the buttons actually are
Select your VM in the left tree, then look at the top-right corner of the content area. There’s a button labeled More. Click it and you’ll see Suspend and Resume in the dropdown. That’s it.
If the VM is running, Suspend is available. If it’s suspended, Resume shows up. The buttons are context-aware, so you won’t see both at once.
What suspend actually does
Suspend writes the VM’s RAM to disk and stops the VM process. It’s not a clean shutdown—the OS inside doesn’t get a chance to flush anything. When you resume, the VM picks up exactly where it left off, but if the disk state changed underneath it (like a snapshot rollback), you can get filesystem corruption.
For Linux VMs, suspend is usually fine. For Windows, it can be hit or miss depending on the virtio drivers. Test it on a non-critical VM first.

Why hibernate isn’t there
Hibernate is a guest OS feature, not a hypervisor feature. Proxmox can’t tell Windows to hibernate—it can only suspend the VM at the hypervisor level. If you need hibernate, you’d have to trigger it from inside the guest, and that’s not exposed in the Proxmox UI.
Some people confuse hibernate with suspend. Suspend is the hypervisor pausing the VM. Hibernate is the guest OS writing its own state to disk and powering off. Proxmox only does the first one.
CLI alternative
You can also do this from the shell:
qm suspend 100
qm resume 100
Replace 100 with your VM ID. That’s handy if the web UI is being slow or you’re scripting things.
My take
The buttons have been there for years, but the feature request thread still claims they’re missing. That’s because people look for a dedicated Suspend button next to Start and Shutdown, and it’s not there—it’s in the More dropdown. Bad UI placement, but the functionality exists.
If you’re doing this often, just use the CLI. It’s faster and you can script it. For occasional use, the dropdown is fine.
Related: error:0A000126:SSL on Proxmox Web Interface: Fix the Unexpected EOF