How can I increase my disk storage without rebooting?

You wouldn't be able to resize your root filesystem without shutting down your Linode first. If a brief outage is acceptable, and if you have space which can be allocated towards your disks, so you can shut down your Linode, resize your disk, and boot it. If you do not have space and you wanted to resize your root filesystem, you would need to resize your Linode to the next plan.

One option that you can take to free up some space is the Block Storage service. Block Storage Volumes can be attached to your Linode while it is running, so you can use it without restarting your Linode. This wouldn't solve the issue by itself, though; you will need to move over any files that do not need to be on your root disk.