Creating panorama in photoshop, can I send that raw file to Lightroom?

Why bother with lightroom at all? You are already in Photoshop, so just take the composited pano and bring it into Camera Raw as a plug in. Gets you the same options (and then some) that you would get with lightroom without the hassles of jumping from program to program.

The right answer for this depends on what your goal is, because there's more than one way to do it and some of them are less complicated.

Prudente  wrote The title says it all, am I able to create a panorama using bridge into photoshop with the merge tool, then have that raw image be sent to Lightroom to be edited? Or does the entire process essentially go from Bridge to Photoshop?

Is there any reason you can't merge the panorama in Lightroom? (Assuming you mean Lightroom Classic.) Is there an intermediate step that you need to perform in Photoshop?

Because if you don't actually need Photoshop in the middle of the workflow, there are two alternatives that are simpler and they keep the resulting panorama in raw format:

A. Select the images in Bridge, open them in Adobe Camera Raw (File > Open in Camera Raw), use Camera Raw to merge them into a panorama, and have Lightroom add the resulting DNG (raw) format panorama.

B. Add the images to Lightroom, select them, and have Lightroom merge them into a DNG panorama. Now when you want to edit them in Lightroom, they're already there. This is the simplest and fastest way because it's all in one application.

I'm suggesting those because the title of your post says "raw file," so it sounds like you'd like to keep the panorama in raw format for further editing. But if you have used the Bridge command Tools > Photoshop > Photomerge, that's the one method that will not preserve it as a raw file. The A and B options above do preserve the panorama as a raw file.

Lightroom and Camera Raw use the same method to merge images into a DNG panorama; it is different than the code Photoshop uses to merge a panorama.

Prudente  wrote Or does the entire process essentially go from Bridge to Photoshop?

The process goes from raw file, processed in ACR/Lightroom, into a rendered RGB file in Photoshop. Then the file has turned into something else, and the process cannot be reversed any more than you can unbake a bread.

What you later do with the RGB file in Lightroom is another matter and irrelevant in this context. It remains an RGB file from the point it leaves initial processing.

Bridge is just a file browser, but it can host ACR the same way that Photoshop can. Bridge doesn't process anything by itself.

if you're starting off from bridge as your first app, just save it in TIFF after you're done from photoshop as Silkrooster said and make sure you leave it 32bit to still have raw capability of edit. Then import the save .tiff into lightroom either by dragging on via the import folder

The image will go back to lightroom when you are done. Just remember when transferring between lightroom and photoshop it is either a tiff, or psd it is never a raw file.