If you can't find gummed brown tape anywhere (ask your local art store, try an on-line art stores, or buy direct), you can use paper that's a heavier weight (thicker) as this is less likely to buckle when it becomes damp and so doesn't need stretching. Try a range of different weights to see what's the minimum you need for your style of painting.
You can also buy watercolor paper in blocks, basically several sheets of watercolor paper on a cardboard base that have been glued together on all four edges so they form a stiff block ideal for using without a board and without needing to stretch the paper. Once you've finished a painting and are sure it's completely dry, you slide a knife in the small gap in the glue on one edge and gently separate off the top sheet from the pad. Peel off the tape if you can, otherwise cut off the edges of the paper or hide them under a mount.
Another option is to staple the edges of the paper into the board instead of using tape. I've never tried it because I tend to be clumsy with a stapler and don't fancy ending up with a board with lots of staples in or having to remove them, but I've heard it works well.