If the satellite's signal is broadcast from the centre of the satellite, so your fixed-radius orbit doesn't alter the distance between you and the antenna, you'll get relativistic doppler shift:
According to "The relativistic Doppler shift in satellite tracking"
Navigation Technology Satellite 2 carries aboard it two
high‐precision cesium beam atomic standards.[...] This observed bias
was 4.47 parts in 10^10, which is consistent with a predicted
frequency bias arising from the special relativity time dilation
effect coupled with a frequency blue shift predicted by general
I would describe that as a relativity-induced error of 0.447 parts per billion
On the other hand, if the antenna was on the satellite's surface, you'd also see doppler shift as the distance between you and the antenna varies.
According to "High-Accuracy Prediction and Measurement of Lunar Echoes"
The relevant rates of change are usually dominated by Earth rotation, which at the equator amounts to about 460 m/s. As a consequence, two-way Doppler shifts can be as large as ± 440 Hz at 144 MHz, ± 4 kHz at 1296 MHz, and ± 30 kHz at 10 GHz.
So, if your satellite is the size of earth, you might see a doppler shift varying between +3000 parts per billion and -3000 parts per billion over the course of an orbit.
However, communication standards already expect a certain amount of doppler shift - according to "Next Generation Wireless LANs: 802.11n and 802.11ac"
in 802.11n [...] the values for Doppler spread in the 5GHz band is approximately 6Hz
So the makers of the standard already expect a doppler shift of 1.2 parts per billion. Cell phone mobile data standards like 5G worry about doppler even more - allowing them to work even on high-speed trains.
But meanwhile, according to "Clock Solutions for WiFi (IEEE 802.11)" (admittedly a dated source)
The typical performance requirement is ±25ppM all-inclusive frequency
stability (includes initial calibration tolerance at 25°C and
frequency changes over operating temperature, power and load
fluctuations, and aging).
So WiFi hardware makers are already compensating for a 25,000 parts per billion crystal frequency error, which can vary with changing temperature conditions.
I would say, as long as any motion-induced doppler frequency changes are small and slow compared to the thermally-induced crystal frequency changes wifi can already account for, you should get normal performance from a frequency perspective.
Of course, depending on your orbital radius you might encounter problems due to signal latency - and inverse-square-law signal attenuation. And the signal attenuation happens in both directions, so even if the satellite has a megawatt transmitter to give a really long range, you won't get a two-way link unless your wifi hotspot has the same great range.