Unfortunately embedding fonts in PSD files is not possible at this moment.
It was already suggested a few years ago on the Adobe Photoshop forums, for example here.
The Photoshop file format does not support embedding fonts. It preserves the fonts appearance, but it doesn't allow editing the text layers, if you don't have the required font installed on your computer (they'll be replaced with another font). You could save the file as Photoshop PDF in order to embed the fonts, but it won't be editable either on a computer that doesn't have those font installed.
Actually the Photoshop format is so bad that in some cases it doesn't allow editing text layers if you didn't have the specific font installed in the first place. I just ran a little test: I created a Photoshop PDF file with a custom font text layer. The font was embedded (checked in Adobe Acrobat). Then I uninstalled the font. When I tried to edit the file in Photoshop it prompted me to change the font (since I didn't have the one I used to create the text layers). I closed the file and restored the font. But when I reopened my file, Photoshop still prompted to change the font - for some reason it didn't recognize the font, even though it was there and I was able to use it.
So, if you want to edit the PSD file on another computer (or by someone else), you've got few options (none of which would be a perfect solution for you, I guess):
- Use common fonts, such as Arial, Tahoma, Verdana, etc. (this is the simplest but probably the worst solution);
- Include the font files manually (you should check the font's license terms to find out if it's permitted). Due to some strange reasons mentioned above, it's not guaranteed to work on another computer, but you can always "change" the font to the one used originally and preserve the file appearance;
- Rasterize text layers if they don't have to be edited later (you might want to duplicate the text layer first and hide the copy - just in case);
- Convert text layers to shapes - it's better than rasterizing, because it allows you to change it's size and shape, without losing the quality. It doesn't allow editing text though, so it's a good idea to duplicate the layer before converting it.