Does Radio Communication Have to Account for the Doppler Effect?
Does radio communication have to account for the doppler effect?

Yes.
Would be pretty terrible if that wasn't the case: RADAR wouldn't work!

Do moving objects such as planes and rockets have to account for this,

Yes. Phones in cars and trains, too. Your 5G NR phone of the future operating above 60 GHz, will have to do that at walking speeds, too.

or is this movement negligible compared to the speed of light?

You can calculate that yourself! With $f_0$ being the original frequency of your signal, and $c_0=3\cdot10^8\,\frac{\text{m}}{\text{s}}$ the speed of light, the Doppler shift as seen by an object travelling at speed $v$ is $\Delta f= f_0\cdot\frac{v}{c_0}$.
So, no, for most microwave communication (that's basically everything &quot;broadband&quot; that you do these days – TV, cellular, Wifi, drone video downlinks, the SpaceX video stream from a rocket), that's not negligible, when you move at high speeds.
And really, 200 km/h is pretty high speed, for example, for typical cellular comms (phones!). Say, your phone is using a band at 1.8 GHz, and you're sitting in a car going 100 km/h = 28 m/s. Then, your Doppler shift is 1.8·10⁹ Hz · 28/3 · 10⁻⁸ = 168 Hz. That doesn't sound so much, but it means that it takes but 1.5 ms until your phase has rotated by 90°, if my in-head calculations aren't off.
Since high-rate communications usually needs the phase to be consistent, you need to measure your channel way more often than that – so, more than 1000 times a second, your channel changes due to Doppler, if you don't correct the Doppler shift first!
And that, at a relatively benign 100 km/h (LTE does 300 km/h of bullet trains), at a relatively benign 1.8 GHz (modern WiFi works at 5.8 GHz).

How about communications with high velocity spacecraft and probes on other planets?

Of course. When you want to receive what was transmitted by a low-earth orbit satellite (the lower a satellite is, the faster it has to orbit around earth), you first have to correct for Doppler, before you can do much else.