Why don't galaxies orbit each other?

There are plenty of satellite galaxies orbiting larger galaxies. The question is how long are you willing to wait for an orbit?

The Milky Way has a mass $M$ of something like $6\times10^{11}$ solar masses, or $10^{42}\ \mathrm{kg}$. The small Magellanic Cloud is at a distance $R$ of $2\times10^5$ light years, or $2\times10^{21}\ \mathrm{m}$. A test mass orbiting a mass $M$ at a separation $R$ will have a period of $$ P = 2\pi \sqrt{\frac{R^3}{GM}} = \text{2 billion years}. $$ Such a system could undergo at most $7$ orbits in the entire history of the universe. The universe isn't old enough for the nearest major galaxy to have completed a single orbit around us at its current separation.

Even if you did wait long enough, galaxies aren't particularly good at holding their shape. If you put them in a situation where gravity is strong enough to bend their path into a closed orbit, odds are they will also be tidally torn apart by that same gravity. And we see this all the time, as for example with the Mice Galaxies:

They do! There's an entire class of galaxy, called a 'satellite galaxy' which is defined entirely based on them orbiting a larger galaxy (which would be called a 'central galaxy'). Our own milky-way is known to have many orbiting satellite galaxies, or at least 'dwarf-galaxies'. If dwarf-galaxies aren't enough, the milky-way itself is gravitationally bound to the andromeda galaxy, and they are effectively orbitting eachother. Because of the tremendous size-scales, however, the orbital period is billions of years --- in many cases, far longer than the age of the universe, so that a pair like the milky-way---andromeda 'local group' actually hasn't completed a single complete-orbit in the history of the universe. That's why we can definitely never (even hope to) see galaxies orbit in real-time.