Yes. Go to File > New and put the background as transparent, and the proportions as the same as the original image.
Using the "Rectangular Marquee Tool", select the whole original image, copy it, and paste it into the "new" image (blank one with a transparent background).
Use the magic wand, and select the yellow background. Make sure nothing but "Contiguous" is ticked. Adjust the "Tolerence" setting to however much needed, until most of the yellow is selected. Then, hit the delete button. If it doesn't show grey and white squares where the yellow was, then you have done something wrong or I haven't explained it good enough.
Go to File > Save As and select it as a "CompuServe GIF", or a "PNG". I'm sure the former preserves the transparency, but I'm not sure about the latter.
When doing the settings, set the colours to how you want