English: I can has cheezburger?
French: Je peux a tchizzeburger?
German: Ich kann hat Käzburger?
Icelandic: Ég get átti óstborgari?
Interlingua: Io pote ha caseoburger?
Hungarian: Sajtburgert adhatat nekem?
Carry on…
Spanish*: ?Puedo tiene quezoburguesa?
*IANA native Spanish speaker.
I would say “?Puedo agarrar hamburguesa con queso?”
It’s supposed to be somewhat gramatically incorrect. I just wish I could enable an option on IE to type in Spanish so I could add the reverse question mark. I can’t use Firefox 'cause my computer refuses to run it for some reason.
Right - that’s why I conjugated “tener” for 3rd person singular (or meant to). That parallels the English:
I can = Puedo
(he/she/it) has = (el/ella) tiene
Would you go for “agarra”?
Swedish: Jag kan har ostburgare? Eh, it just sounds awkward. I’d say “ostmacka” - cheese sandwich - because it has the same cadence as the original sentence. JAG KAN HAR OSTMACKA!
Much better.
AY NO!
Do not agarrar la hamburguesa… do not coger la hamburguesa… do not pillar la hamburguesa… que no… tener. Promise. So far, I haven’t met any Hispanic who considered “tener” a synonim of a certain English word which begins with fu and ends with ck.
“Hamburguesa con queso” is the real name of a cheeseburguer, but I’ve seen them called quesoburguesa in the OP’s context. I think Heart’s “¿poder tener quesoburguesa?” wins, with the grammar change.
Using the infinitive for every verb has a pretty long story behind it in Spanish as a way to show that someone knows the vocabulary but not the grammar, it even has a specific name: “hablar como los indios,” “to speak like the ‘Injuns’,” because where a Western had them speaking Spanish in the original, the dubbed version would have them speaking in infinitives.
Cantonese: Dak yau chi si bou bau ah?
!Que fascinante!
Russian: Магу ли я палучить чисбургор? [Magu li ya paluchit’ chisburgor?] (standard Russian would be Mozhno mne chizburger?)
Russian has a lolspeak equivalent (even more widely used) called zhargon padonkov/olbansky yazyk (scum slang/Olbanian)
…it should probably be “jag kan får ostburgare”.
sneaks out again
Esperanto: Mi povas havas fromazburgo?
(The correct version would be, Cxu mi povus havi fromagxburgon?)
Am I the only one who read that as shitburger?
Rather: Ég get fád ostborgari
Well, it’s supposed to be grammatically incorrect with spelling errors, hence “óstborgari” instead of “ostborgari”… I don’t see why “fád” is any better than “átti” though.
I don’t have anything to add, just wanted to say I laughed out loud when I read your post and that’s not something that happens to me very often. Thanks! 
Norwegian: Jeg kan får cheezburger?
(Norwegians say “cheeseburger”. Sorry. If you really want it to sound foreign, you could say “osteburger”, I suppose, but the only people who use that word are about nine very conservative Norwegian language professors who spend most of their time telling everyone else that they talk wrong.)
Bulgarian: Аз може ли да има хамургер с сирене? [az mozhe li da ima hamburger s sirene?]
átti would make the sentence mean “I had a cheeseburger” while fád makes it “Can I have a cheeseburger” - which is more correct.
Tarentino
French: Je peux a royal with cheese mutherfucker?
Irish:
Ar feidir liom cheezburger a bheith agam?