Enlarging an image happens one of two ways: Either by using a vector-generated graphic, or by lowering the resolution. Scans are not vector-generated, so if you want to enlarge a scan, you should scan at a higher resolution than you need for the quality of output you want, and then set the output for the size that you want. So if start with an image at 100% and you want the image to be 300 dpi or better at 150%, your scan should be at least 450 dpi.
For a stained glass project, the resolution probably does not need to be so high, and most scanners will scan at a high enough resolution that the results will be adequate unless you are multiplying the size quite a bit.
Your biggest problem may be that if you start with a letter-sized image, you may need a large-format printer to print the result.
If your printer can handle it, all you need to do is to print the image at 150% (to use your example) to enlarge it. Some all-in-ones can do that sort of enlargement when they copy, so no computer would be necessary.