Yes, you can chain up multiple um-zu clauses without a problem. Obviously, it's not as short and elegant as in English though. You're repeating um and zu, which can make it sound a bit clumsy. It might be better style to avoid this, but there's no grammatical problem with it.
About your translation: you decided to make "jemanden" an object of "Ich suche" while it is an object of "to meet" in English. This leaves "treffen" without an object, which doesn't work.
"I am looking to [do something]" can be translated as "Ich würde gerne [etwas tun]" or "Ich beabsichtige, [etwas zu tun]".
I would translate it as:
Ich würde gern jemanden persönlich treffen, um Deutsch zu üben.
Ich würde mich gern mit jemandem persönlich treffen, um Deutsch zu üben.
If you'd rather use "jemanden suchen", you can do that, but then you run into a little gendering issue with the pronoun you need as an object for treffen:
Ich suche jemanden, um sie oder ihn persönlich zu treffen, um Deutsch zu üben.
You could use "sich [mit jemandem] treffen" instead of "jemanden treffen" to avoid that. (It's slightly colloquial to leave out the "mit ..." object here.)
Ich suche jemanden, um sich persönlich zu treffen, um Deutsch zu üben.
You probably also see how "um...zu" is a bit repetitive here. It's fine in my opinion though as long as it's only twice. There are ways to circumvent this, like:
Ich suche eine Person, mit der ich mich treffen kann, um Deutsch zu üben.