How can I tell if the server supports hot-swapping drives?

Grrrr... I hate Supermicro... and white box servers in general - There's no consistency in model numbers or parts. If HP, IBM, Dell, etc., we would be able to look at your server model number and say, "yes, this supports feature X".

For you, this is going to depend on the controller inside the server, as well as how the SAS backplane is connected.

Please provide the information from lspci on the server. Perhaps df -h and lsscsi or cat /proc/scsi/scsi.

This will give us some RAID controller information if (present) and perhaps some information to proceed with.

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Edit:

Your server supports hot-swap.

From: https://github.com/IQAndreas/computers/tree/master/austere-armadillo

This is an older 2007/2008-era server.

Here's your RAID controller's logical drives.

[2:0:0:0]    disk    AMCC     9690SA-8I  DISK  4.08  /dev/sda 
[2:0:1:0]    disk    AMCC     9690SA-8I  DISK  4.08  /dev/sdb 

and

Host: scsi2 Channel: 00 Id: 00 Lun: 00
  Vendor: AMCC     Model: 9690SA-8I  DISK  Rev: 4.08
  Type:   Direct-Access                    ANSI  SCSI revision: 05
Host: scsi2 Channel: 00 Id: 01 Lun: 00
  Vendor: AMCC     Model: 9690SA-8I  DISK  Rev: 4.08
  Type:   Direct-Access                    ANSI  SCSI revision: 05

It's a 3Ware 9690SA-8i, which uses the 3w_9xxx kernel module in Linux. You have a pair of disks in RAID 1 and four disks in RAID 5. These are SATA disks, so I recommend reconfiguring them to RAID 1+0, giving you 2TB of usable space on the larger array. You'll be able to do this from the BIOS by pressing Alt-3 during the RAID controller's initialization. The manual for the RAID controller is here.