The DM and player can decide this together, and it'll differ based on the character of your game

Rules as written, there is no requirement to wait, or to be anywhere in particular, when the companion is granted. (This is true for abilities granted at levelling up in general, e.g. knowing new spells, or having an ability you didn't have yesterday.)

In more "gamey" campaigns it's fine to just handwave this and say "Poof! Now you have a camel!"

In a more story-based campaign, think through how you want this to add to the story. Since it's supposed to be "free" for the character I wouldn't make it onerous, but (especially if the player is amenable) it's fine to make it interesting.

• Perhaps you spend a night meditating and fast in the woods, and it comes to you?
• Perhaps you go out, track it down, and tame it?
• Perhaps the animal was sent by your deity, and meets you at some reasonable location?

In a campaign with a high level of realism, I'd talk with the player about is/n't possible. For example, at this level you might rule that certain kinds of companions that are normally allowed aren't realistic to find nearby in your biome.

The Falconer page states:

The bird can be of any type of large hunting or scavenging bird (even a vulture). This ability functions like the druid animal companion ability (which is part of the nature bond class feature), but the falconer must take the bird animal companion, and that companion has only half the normal hit points.

Since the system doesn't specify exactly what qualifies as a bird of prey aside from telling you it can be any kind of large hunting or scavenging bird but that it does allow you to take a vulture, The likely course of action would be that the Ranger would have access to any Bird type animal companion. So you could take an Eagle, a Falcon, a Roc, or a vulture, provided it had the Bird sub-type.

At 4th level when Ranger would normally get an animal companion, instead your animal companion would level up and would have 100% of its HP instead of the half it starts off with.