I looked for a phoenix in the Monster Manual, but I can't seem to find anything. I want to see if I can use one in my adventure, but I couldn't find a phoenix anywhere.
Are there phoenixes in D&D 5E?
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There is a creature in D&D 5e called Phoenix. It's in Mordenkainen's Tome of Foes (p. 199) and is a CR 16 Gargantuan elemental.
There's also the Arclight Phoenix from Guildmaster's Guide to Ravnica (p. 193) which is a more moderate CR 12 (and only Medium sized).
If you want even more modest phoenixes you'll have to get cobbling. Fortunately, phoenixes are an example in the Dungeon Master's Guide (p. 273) for modifying a creature:
Need a fiery phoenix? Take the giant eagle or roc, give it immunity to fire, and allow it to deal fire damage with its attacks.
You could even base it on an even smaller/weaker bird, say if you want it as the familiar for a magic school's headmaster.
As a general method, D&D Beyond's Monster Listing is good for finding whether official stat blocks exist, and will tell you where it can be found, even when you can't access it on that site.