That's not to say all is lost, but your environment is in a precarious state. VMware will maintain Technical Guidance for vSphere 5.5 until September 19th, 2020. Support requests can only be opened via the self service portal for severity 2 and lower issues, so if you experience an outage, you are on your own.
These tickets only apply to supported configurations as well. As a VMware Support and Subscription (SnS) customer, you need to upgrade before September 19th to avoid losing the full protection of a supported vSphere platform. 'You're right, we need to upgrade. But what does that all entail?' The vSphere platform has a very specific upgrade path you need to follow to ensure no service interruptions during the process. Before we dive into the VMware aspects, there are other variables to consider.
Let's start with the host servers; the hardware ESXi calls home. Validate that the hardware is compatible with the version of ESXi you are upgrading to via the. Take a look at host as well. The host hardware vendor may also provided information about compatible versions of ESXi on their website. Even in the event that the existing BIOS and firmware is compatible with ESXi 6.0 or higher, now is a great time to upgrade. This helps to keep your hardware secure and up to the manufacturers recommended levels. One of the first things the vendor's tech support engineer is going to tell you to do is to upgrade the BIOS to troubleshoot any problems you may call in with.