As long as you don't mind using a simple tea strainer on your cup or serving pitcher, you should be fine with a gaiwan. You can always put the leaves that got held up by the strainer back into the gaiwan for the next infusion.
A nice handmade Japanese kyusu will give you a more enjoyable tea making experience, of course. The tea itself, however, should turn out just fine brewed in a gaiwan.
A good starting point for brewing (sencha) is 70° C for 1 minute. Adjust to taste. Second infusion 5-15 seconds, slightly higher temp. Adjust to taste.
Increase time and temperature for further infusions.
Use at least 0.5 g of tea per oz of water. And... adjust to taste