How is my sentence? 我以为我没有工商卡，可是，我顺便找到了。
It sounds weird using 顺便 here. I think you should use 无意中 (by accident/undesignedly) instead, e.g. 我以为我没有工商卡，可是（or use 结果 instead to emphasize the opposite result）我无意中找出来一张。

Note that 顺便 is intentional. e.g. 今天我去见我导师，顺便去大学图书馆借本书。The action that going to the library by the way is still intentional, it's planned. But for this case, the result that finding out the card is unmeant, undesigned. As you said 我以为我没有工商卡, so you won't try to search the card intentionally.

PS: 工商卡 is valid, but we use 工行卡 or 工行的卡 more because 工商银行 is abbreviated to 工行 ususally. Similarly, you can say 招行/建行/中行/农行 etc.