Answer posted under the wire during closure. Others may be able to write an additional and/or different answer as soon as two more reopen votes are cast.
Would it have been possible to send a radio signal towards ʻOumuamua?
Yes. It is certainly true that at any time, a large radio telescope or even a small transmitter can always sent a signal toward something. New Horizons can receive our signals now even though it's in the Kuiper belt because it knows when and where to listen and at what frequency.
As we could not get any radio signals from asteroid ʻOumuamua, couldn't we have sent a powerful radio signal to it and then check if we can get any radio signal in response?
The asteroid was quite close to us in the 2nd half of 2017, but now it's much farther than New Horizons. However if it had advanced technology it would know we were very active in the electromagnetic radio spectrum and it might be listening. If so, we certainly could have sent it a signal that it could receive.
Checking for a response is also hard because we'd need to know what frequency and when. We'd have to allocate resources, etc.
For the frequency, see also
Below: Plots of the distance of ʻOumuamua (bumpy, blue) and the New Horizons spacecraft (straight, green) from Earth as a function of years since 2017-01-01.