No. You're going to have to download a program. If you want a program that works, and you can save decent JPG's from, and format accordingly to Tapastic's requirements, you're going to need to download something. I'm not saying that because there aren't browser-based programs for it - I'm sure there are - but because it'll help you better in the long run to just download a program, so that you can at least archive your files, keep track of them, and not have to worry about lack of internet access, among other things.
It doesn't have to be a paid program. You can get a program called Fire Alpaca, which I've heard is like a less complicated version of Photoshop, and free. There's also GIMP, which is pretty simple to use. And you can change the resolution and canvas size to get the 8-bit look.
Paint Tool SAI is about $70, if you ever run into any money. There are other ways to getting it of course without shelling out the money, but I'm not going to disclose said methods :B It even has a binary tool which is used for pixel-looking art, including 8-bit and 16-bit.
So yeah, my recommendations would be to try out Fire Alpaca or GIMP, and if you ever get the money/figure out how to download it, get SAI. SAI is my personal favorite, but I've heard good things about Fire Alpaca.