The way to politely ask for things in Icelandic is with “Can I get ...?”. The word for “to get” is “að fá” but the word “can” makes it radically alter its form (see the article on the past participle for the reason why) so the result is “Get ég fengið ...? Fá and fengið are the same word, just with a different form. Click here to see the full table of conjugation for this word.
- Get ég fengið hamborgara? (Can I get a hamburger?)
- Get ég fengið kaffi? (Can I get some coffee?)
- Get ég fengið fisk? (Can I get fish?)
- Get ég fengið tvo bjóra? (Can I get two beers?)
The word you’re asking for has to be placed into the accusative case (the second case).
English speakers have a tendency to incorrectly ask for things with ”má ég fá ...?“ because they are directly translating the English phrase ”may I get ...?“. But ”má ég“ can usually not be used in this context, ”má ég“ means ”do I have permission to" and so is mostly used to ask permission from an authority figure. For example, you might use it to ask for permission from your parents.
- Mamma, má ég fá nammi? (Mom, am I allowed to get some candy?)
- Mamma, get ég fengið nammi? (Mom, could you please hand me some candy?)