That’s not clear in the description of the spell in the phb. Is that rule clarified elsewhere?
"Choose one of the following options". It certainly does not say they can choose the particular animal.
In my games, if the druid summoned some vultures to shuttle the PCs across a gorge, I'd make them do a few easy animal handling rolls
Animal handling is used for many purposes. Most of them focus on what you can extract from the mount - intentions... behaviors ... However, some of them are whether YOU can interact with the mount correctly physically - such as when you do a risky maneuver.Just as a point of order, they aren't vultures, they are Fey in the form of vultures that follow verbal orders. No roll should be necessary.
The text is ambiguous. Both "player chooses the creatures" and "player chooses the CR, DM chooses the creatures" are consistent with the spell as written. However, normally the caster makes all choices about a spell... and the DM has enough to do without combing the Monster Manual for suitable critters mid-combat. So, I favor the "player chooses" reading.