While it would make sense, there would be some tricky parts to it...
As one might noticed, she doesn't have a lot of Westerosi folks in her ensemble. Jorah is gone, and it is just Barristan who remains but was kicked out of the KG and placed in self imposed exile. Sending someone who is a complete foreigner in Westeros might be difficult, with the language barriers, cultural confusions, and the fact that a person who is suppose to be a spy ends up sticking out like a sore thumb because of their different appearance, attire and manner of speech, and education. Especially since Dany doesn't have a lot of "Highborn" people, mainly former slaves, servants, children and warriors. The only Highborns she knows that could somehow wiggle their way into a place like KL with composure and court knowledge, are the ones in her current city, including her curernt husband, and they wouldn't be trustworthy enough to go on her behalf.
Where could her informants go with that in mind too? How could someone like Jiqhui, her bloodriders, Strong Belwas, her Unsullied, merceneries, peasants, sneak into a significant place somehow? And as mentioned, Barristan can't really go back to KL himself, he is already well known, he is outcasted and shamed. At most he could try and be a Kingsguard again under Tommen. But still, he is needed by Dany's side, training upcoming knights and he himself might feel too old to do such a deed. Who could survive going to the wall, or the North in general? Storm's End? Where the Ironborn live? Dorne will be out of the picture now after Quentyn's sizzling moment. There is Missandei, who is brave, intelligent and linguistic, but Dany might not want her away from her side. Missandei also might be too exotic and young looking to probably blend in anywhere, unless she works with someone else under a plan.
She doesn't have someone like Varys or Littlefinger, nor would she have the time to raise people up like that. Varys and LF were originally commoners, working their way up to better status and placing themselves in a noble world, but that took a long while for it to happen. Varys little birds are all local kids spying on local locations.
There is Daario, but how reliable would Daario be? And he is also mainly busy with leading his own men.
Alongside all that too, many of her Dothraki still fear the water. Imagine trying to send one of them alone across the sea, with only a fear companions! Instead of at least being together as a large group. And even if they did make it, they would be considered enemies. Once someone finds out they are Dothraki, trouble would arise.
At most, she probably should have asked more questions about Westeros with people like Jorah, Barristan and Qyentyn. Barristan and Qyentyn especially, since Barristan can give insight whatnots of how courtlife was, about the last two kings he served personally, and Qyentyn about the specifics of Dorne and political situations and the war. It wouldn't be a mind boggling amount of information, but it would still be the important enough basics, and she could still use it to prepare in some way.
If she were to choose informant(s), it would be a long decision and likely some training involved.
And then is of course, possibly Tyrion, if they ever meet.
Then there is obviously the geographical issues, the time it would take to get information and the risk of death for the informant. Plus, who knows, maybe Dany doesn't feel comfortable going that route. She was the target of informants, and that felt awful. And she is likely a target of informants once again in Meereen. She isn't really in a trustful mood over things like these. Having informations everywhere, ones for her and ones against her, might be too much. Perhaps she would worry whether they would easily get bought out too, if not killed by whatever enemy it might be.
But altogether, yeah, I definitely agree that Dany really does need to do more to educate herself on places like Westeros. Come on, girl!