Does every OS need RAM?

Does every OS need RAM?
For IBM PC compatible hardware, a mandatory step of the BIOS POST process is to check if there is RAM to load the BIOS into. Optionally the POST process checks of your RAM functions correctly. After the POST process, the BIOS loads the bootloader to the RAM and gives control to the bootloader. So the answer to your question ("Does every OS need RAM?") is: yes, every IBM PC compatible hardware requires at least some functioning RAM to boot. This is true for any OS that runs on that hardware.

Note that in the OP's original question, there was a reference to a "laptop", which I interpreted as: IBM PC compatible hardware. For the remainder of this answer, I'll assume IBM PC compatible hardware.

Can an OS boot with faulty RAM?
If your RAM is faulty (and not entirely absent/broken) or if you can (partially) replace your RAM, you might be able to boot using the BadRAM of BadMEM kernel patches. It requires you to recompile the kernel (sounds easier than it is if you do it for the first time) and you can reboot and tell the kernel where your bad memory is. A nice explanation of the use of Memtest86/Memtest86+, BadRAM/BadMEM can be found here.

Can an OS boot without RAM and using the CPU's cache as RAM?
As far as I know there is no way to use your CPU's cache as RAM without any RAM present on your system (as is suggested by @philipp and other) in the comments. If there is, it would be nice to add it here. The only paper I could find on this subject is this paper that states: "Using processor’s cache as RAM until the RAM is initialized". Not sure if (and how) it will work without RAM. As far as I know there is no working code that boots an OS on a IBM compatible PC. Any references to proof-of-concepts, working code or anything is welcome in the comments and I'll add it to this answer.

Can I get to the BIOS?
The OP's question is a bit vague if the laptop is able to pass the BIOS POST. As @Tonny points out, no OS will help you to "get to the BIOS". You enter the BIOS using the F1 or F2 or F10 or DEL or ESC key, depending on your brand of BIOS.

How to recover data from the laptop without RAM?
As for the goal behind your question: why do you need access to your laptop? Probably because there is still data on the HDD that you want to recover? If that's the case, it is far more easy to pull out the HDD (see manual) and attach it to an external storage device or directly to a PC. Here is a nice guide to do just that.

It's possible in theory, but it would be very slow, since it'd need to use the disk for any temporary storage that didn't fit in the CPU caches. (CPUs have a few megabytes of cache because even RAM is too slow for them. Think about that.) So you'd need a rather small OS.

(Well, embedded systems-on-chip can execute code from either RAM or internal Flash memory – the PC BIOS could in theory do the same, but it can't run the whole OS.)

But, another thing is, the BIOS runs before and outside any installed operating system, and it's the BIOS that lets you boot from a pendrive or whatever. So if you cannot reach the BIOS, then no other OS in the world is going to help you.

You could, with some effort, design a system that contained no RAM. Load your software from ROM (or storage) and do everything in registers or on cache. Such a system would have exceptionally narrow use and given today's RAM prices be a bit pointless. An off-the-shelf laptop will not function without some onboard memory.

Your real question is more likely "How do I get information off of a computer that will not boot" and that's easy. Disassemble it, remove the drive, and connect it to an external drive case.

Note, however, that any computer old enough that you can't just get $20 worth of working RAM for it will have an ATA ( regular ATA, not SATA ) drive. This interface is nearly extinct today, so you will also need to find a drive case with an ATA card as well as a full-size to laptop-size adapter. I have one that I bought in 2000, and I keep a couple of old cases around just for this purpose.

If I read the question correctly everybody here is barking up the wrong tree.

He states explicitly "in order to get to the BIOS".

If the laptop is so broken it won't even get into BIOS, everything else is pointless.

You can't boot ANY OS on this regardless of the RAM situation.