Learn Swedish: Vi skulle vilja ha notan, tack. - We would like to have the bill, please.

Yes, it is! But just like in English where 'will' becomes 'would' to be polite, Swedish uses skulle to form a soft, polite request. This is called the conditional. Saying Vi skulle vilja ha... (We would like to have...) is much more polite than saying Vi vill ha... (We want...).

Can I leave out ha and just say Vi skulle vilja notan, like the English 'We would like the bill'?

Not quite. In English, you can say 'would like' followed directly by a noun. In Swedish, you generally need a verb to explain what you want to do with the noun. So you must include ha and say skulle vilja ha (would like to have) before the noun.

Why are both vilja and ha in their infinitive (dictionary) forms here?

Because they follow the auxiliary (helping) verb skulle. In Swedish, the first verb in a chain gets conjugated (here, skulle), and any verbs that follow directly in the same chain stay in their basic infinitive form. So the structure is skulle + vilja + ha.

Why is it notan and not just nota?