There's a simple problem here and a more complex problem.
If your list of symbols is static, then you could use any typical autocomplete API against a file that you maintain locally.
However, the list of symbols is rarely static. Symbols are constantly being added or erased and changed due to a variety of financial market situations (IPO, acquisitions, mergers, renames, bankrupcies, etc.). Many symbols are also traded on specific exchanges, and some symbols are cross listed. There are also financial instruments that are not simply stocks, such as indices, commodities, etc. The general term for this is "instruments definitions" and a complete list of such definitions is a service provided by companies such as Reuters or Bloomberg.
I am not familiar with any free and open instrument lists that you could get for free, and you need to make sure that you are complying with the licenses of services that allow you to get a current
list.
If you can tolerate a delay of one business day, you might be able to scrape the list from a variety of sources that provide close-of-business listings for all stocks in the US. The WSJ has a printed list (likely also an electronic one). Eoddata provides such a list, etc. But make sure that you are complying with their terms.