I was visiting the Louvre recently, where I tried to ignore the lines to Leonardo's Mona-Lisa and looked at all the other great art instead, such as Michelangelo's dying slave and the many works by Raphael on display. I somehow managed to miss or not notice Donatello's works in the huge museum, but I know that there are several of them as well. Nevertheless, this visit sparked the obvious silly question:

Which museums have artworks by all four ninja turtles?

Combined of course with my childish desire to visit all of them. The two I found so far are the two obvious places I've already been to:

• The Louvre


• The Vatican

Are there any others? The next obvious place to look might be Florence, but while there is a lot of each of them, the closest I could find was the Uffizi gallery, which seems to be missing Donatello.

It seems that the National Gallery of Art in Washington DC has works by all four artists in their permanent collections, but for Michelangelo they may only have some simple drawings. I don't know if they are all on display though.

I came to this conclusion by using the website artsy.net. There you can look up individual artist and it will give a list of artworks by them, which can be sorted by institution. I did this for all four of them and there were several who had Leonardo, Michelangelo and Rapahel, but Donatello seemed to be the "bottleneck". Here is what I found for the National Gallery of Art:

• Leonardo


• Michelangelo


• Raphael


• Donatello

All works by all artists are of course not listed on this website, so there may very well be other institutions that also have works by all four artists.

Update:

After reading @Pere's answer, I realized it might be possible to find information about which works are on display on the museum's website. Here is what I found:

• 
  Leonardo da Vinci: Two paintings are on display.


• 
  Michelangelo: Non of his drawings are on display. A sculpture by one of his followers is on display.


• 
  Raphael: Five paintings on display.


• 
  Donatello: Strangely, they don't list any original in their collection. The ones on arty.net are not listed at all. They have two copies and two works by followers on display.

So you probably cannot visit the museum and see originals by all four artists on display.

The Victoria and Albert Museum in London has works for all 4 turtles in its catalog, in particular the Raphael cartoons. There are also some notebooks from Leonardo, and a drawing by Michelangelo (as well as a number of casts of his sculptures in the Cast Courts). I seem to have temporarily broken their search engine, but this page mentions having pieces by Donatello as well.

Edit: The Renaissance collection has a number of pieces by Donatello, e.g. this relief.

The National Gallery in the UK lists paintings by all four 'turtles' in its Glossary. I don't know if all four are on display though.

-EDIT-

I saw Donatello listed in their glossary but yes as pointed out in the comments it might not be correct that his work is owned/displayed by them. I believe after more research he is mentioned as he inspired works by other artists (that they do own).

For a bonus snippet, while looking at this and depending on your definition of 'artwork by' the V&A in London has work by Raphael, Donatello Leonardo's notebooks and casts of work by Michelangelo.

Raphael—The Miraculous Draught of Fishes, 1515

Replicas of two earlier Davids by Donatello and a number of small scale bronzes.

A full-size replica of Michelangelo's David

According the incomplete data in Wikidata, the only museum with works by the four artists would be the Metropolitan Museum of Arts (still not mentioned in other answers). Donatello and Michelangello have one sculpture each that - judging by their photographs - are on display. One of Rafaello's work is a painting that is on view according to the museum web. Unfortunately, Leonardo's works are just sketches that don't seem to be on display.

Other museums have works of three of them: Albertina, Ermitage, Galleria degli Uffizi, Museu Boijmans Van Beuningen, Budapest Fine Arts Museum, Vatican Museums, National Gallery, and Victoria and Albert Museum. Since data are incomplete, those are likely candidates to have works of all four of them. In fact, some of them have been shown in other answers to have those works.

Note: I don't link a Wikidata query with global results because it took too long to execute and I had to resort to four different querys and mixing their outputs in R.