Am I allowed to look up a card during a game, if it's being played with serious Rules Enforcement? Either to read the full card from the Gatherer, or to try to find the exact card I'm vaguely thinking of.
Two possible scenarios:
An opponent has revealed a card previously, and I want to remember the exact details. (Something like the exact wording of a sorcery, mana cost, keywords on a creature, subtypes, etc. etc.)
I have a "name a card and remove it from the opponent's deck" type card, and I want to, say, ask google for "Blue Gearhulk" to remember that it's that that one is called "Torrential Gearhulk".
Obviously, in casual or semi-casual games neither of these are a problem. Any reasonable opponent will just re-reveal the known card (assuming it IS still guaranteed to be known) or will accept "I name the Blue Gearhulk".
Also obviously, it seems unlikely that I (or anyone else) would get to the stage of playing at the REL without knowing the cards too thoroughly for this to be relevant.
But in principle... is it permitted? Or not? (and if not, is there a good reason, or just "we decided 'No'"?)
You can't lookup things on the Internet during a match per MTR 2.12:
At Competitive and Professional Rules Enforcement Level during drafting, deck construction, and playing of matches, players may not use electronic devices capable of taking and storing notes, communicating with other people, or accessing the internet (with the exception of taking brief personal calls with the opponent’s permission).
As for the Torrential Gearhulk. I've asked a judge at a Standard PPTQ Tournament (Competetive REL) if it's enough to name a card by describing it as "Blue Gearhulk". His answer was that it's OK, because there is exactly one card that can be described as "Blue Gearhulk" in the sets permitted in the format we are playing. This is per MTR 3.6:
A card is considered named in game when a player has provided a description (which may include the name or partial name) that could only apply to one card. Any player or judge realizing a description is still ambiguous must seek further clarification.
MTR 3.6 also clarifies that:
Players have the right to request access to the official wording of a card the can describe. That request will be honored if logistically possible.
So yes, you can ask about the card.
As a supplement to piotrek's answer, there is a story in which Gerard Fabiano repeatedly attempted to name "Humpus Wumpus," for the ability of Demonic Consultation. Humpus Wumpus was not allowable in order to specify Hunted Wumpus. My point is that one judge may take "blue gearhulk" to uniquely describe only one magic card, but a judge may not allow "Humpus Wumpus" to uniquely describe Hunted Wumpus. Meanwhile, they might allow a mispronunciation like "Tarmogoof" in place of Tarmogoyf, and you might get away with naming a legendary creature without enumerating its entire name. A judge might know absolutely for certain what creature you're trying to name, and still not tell you what card it is you're thinking of; similar to Wheel of Fortune* not accepting a mispronunciation, even when all of the letters on the board are filled-in. So a description is acceptable when it is literally correct, rather than commonly clear.
*Originally I had accidentally said The Price is Right instead of Wheel of Fortune.