Most often the result is not fully accessible; browser and assistive technology is inconsistent; and implementations vary across the web. This is why it’s always better to use native selects with HTML/web. (Native components is also a better choice for native apps as well.) Don’t forget that HTML selects can be styled with CSS; see these resources by RTD, Filament Group, and LugoLabs.
If you must implement a custom select dropdown, you will most like need to use the ARIA listbox role, combobox role (which specifies a composite widget), and often a combination of those roles. The option role is also required and usually a few other ARIA attributes (for label, state, etc.).
Here are some great examples which will save many folks a lot of time!
- Anatomy of an Accessible Auto Suggest (UX Mastery) by Adem Cifcioglu
- Accessible Autocomplete by Adina Halter
- Custom Select by Deque Systems
- Combobo — accessible combobox module by Harris Schneiderman (of Deque Systems)
- Added: Building an accessible auto-complete (Slideshare) by @russmaxdesign