The method I use (with Libre Office rather than Open Office, but I doubt that makes a difference) is simply to highlight the block of cells in the spreadsheet, ctrl + c, and then ctrl + v into the body of the email. I can't vouch that it keeps tab stops because I don't use them in a spreadsheet, but it does preserve column alignment, width etc. However you need to make sure that the email is being composed in HTML format, I doubt that it would work with plain text.

I don't see how to compose mail in "HTML format". Where are you finding this option? I tried using "Insert HTML" when pasting into a normal compose window, but the result is a continuous line of text rather than a table.

BTW, the tabs seem to be inserted/deleted by Thunderbird. There's no tabs in a spreadsheet. I've tried lining up the columns with spaces, but I can't make it predictable.

While this image is of a ubuntu linux install the menu in question is same make sure the highlighted option IS checked to get the effect you desire as described in amanchesterman's response.