If you think vegan baked goods are dry and crumbly, you're probably not using the right fat or the right amount of fat. Butter is what makes traditional baked goods so tasty and helps keep them all together, so you can't just eliminate it and expect the same results. Many imbalances in the dough could be preventing your vegan cake from cooking properly. The dough may be too thin or watery; if so, reduce the water content.
It could also be a problem of lack of yeast (baking powder or baking soda), which can be solved by adding more. If the leavening agents are not fresh (that is, they have been open for more than 3 months and have not been stored in airtight containers), you will have to buy new ones. However, there are many specifically vegan cake recipes and tips on how to adapt standard recipes to vegan ones. In vegan baking, egg replacements can often create dense pastries because they lack the egg protein that would normally form the structure of the cake. Also, keep in mind that if you're only trying to replace eggs in a non-vegan recipe, the chances that your baking will fail are high.
Choosing the Right Egg SubstituteChoosing the right egg substitute for a non-vegan recipe, following what the vegan recipe recommends, or sometimes eliminating egg replacements entirely will help avoid a dense result.
Now that you know all the dos and don'ts of baking fluffy vegan cakes and muffins, it's time to preheat the oven and start baking.