Most supermarkets in Australia now sell ingredients such as curry leaves (Indian), galangal (Thai), rice paper (Vietnamese), and radicchio (Italian). Kosher is available, as is Halal. Part of an aisle of a supermarket is usually dedicated to well-priced international foods. There is likely to be a specialist grocery store or market with the food you want in all large Australian cities, Asian and Indian markets are everywhere. Many of these shops also sell spices and other food items online.
Large cities, in particular, usually have a Chinatown where a huge selection of Chinese and Asian ingredients are available. On the edge of Sydney’s Chinatown, is Thainatown which has a number of Thai grocery shops, and restaurants.
It is not only Chinese and Thai communities that have their own shopping districts. Most migrant groups have an area in major cities where you can buy food and cooking ingredients from your home country.