.tif and .tiff are exactly the same file type. Just as .jpg and .jpeg are exactly the same.
As Kappy explained, all you have to do with the TIFF you save is to highlight the file and change the name from doc.tiff to doc.tif. Though it shouldn't make a bit of difference which way you send it. There is literally no difference between the two, other than one letter of the file name. That the company you create these for will only accept files ending in .tif tells me they either aren't very bright, or are using a very old system that doesn't understand four character file extensions.
If I remember correctly, the ability to use a four character extension was added to Windows with Win 98. Could have been Win 95. Anywho, there isn't an easy way to get to a TIFF from Word since it has no option to output a raster file type. It's going to require multiple steps.
The first step is to save the document from Word as a PDF. Office 2011 and 2016 for sure have this option right in its own menus. You can't really do much with the rest of the file types it offers. At least not as easily as a PDF.
The free Adobe Acrobat Reader DC is no help at all. It can't save or export a TIFF. The full version of Acrobat DC isn't helpful, either. I can set it to output a Grayscale, 200 dpi, LZW compressed TIFF, but it does so without any anti-aliasing, and it looks like it. Garbage.
Preview will turn a PDF into a very nice TIFF, but will only do so as an RGB file. You can select 200 dpi and LZW compression for the output, but not a color space. But at least you're partway there. You would then need to process the resulting .tiff file again to change it to Grayscale. You can't do this in Preview. With the converted TIFF open, and then choosing Tools > Assign Profile, it will only let you select other RGB profiles on your drive.
If you have Photoshop, Photoshop Elements, GIMP, or some other raster editor, you can open the TIFF in that app, convert to grayscale, and save it. It's been ages since I used Graphic Converter, but that can likely convert the TIFF from RGB to Grayscale.