If you believe you have been injured by a vaccine, there are two paths to showing that you are entitled to compensation.
For some vaccines, there are injuries which research has shown might have an association between a given vaccine and that injury. There is a list of specific injuries with a potential association to certain vaccines. This list is called the “vaccine injury table” and can be found, along with other useful information about the program, here: https://www.hrsa.gov/vaccine-compensation/index.html.
Any injury listed here is called a "table injury." If you have a table injury, in most cases you would only have to prove that you got the specific vaccine and that the associated injury occurred within the timeframe identified in the table for that vaccine/injury combination. Table injuries do not require evidence that the vaccine caused your injury.
However, if your injury is not a table injury, that does not mean that you are not entitled to compensation from the VICP. It simply means that you must provide evidence that, more likely than not, the vaccine caused your injury. This is usually done with an opinion from a medical expert with the relevant education and experience. Specifically, that expert opinion must show (1) a medical theory causally connecting the vaccination and the injury, (2) a logical sequence of cause and effect showing that the vaccination led to the injury, and (3) a close temporal relationship between the vaccination and the injury.
Whether you had a table injury or not, the process of petitioning the VICP for compensation can be complicated. Therefore, you should contact an attorney with experience handling vaccine injury compensation claims.