Language has been called a Cognitive or Cultural Tool, a description that succinctly summarizes the research on the effects of language on cognition and the mind.
This research also answers your question with a definitive Yes: A broader vocabulary can allow you to think faster. Assigning a label, which is basically what a word is, to a concept allows it to be used more easily in your brain.

Consider the evidence:

One study looked at differences in counting ability and counting memory between speakers of English and speakers of Piraha. The Piraha language is notable for many things, one of which is its lack of any words for numbers or counting. What they found is that the Piraha can still recognize quantities - not surprising. However, their memory for quantities was worse than English speakers'.

Number as a cognitive technology: Evidence from Pirahã language and cognition
Michael C. Frank,Daniel L. Everett, Evelina Fedorenko, Edward Gibson

Dan Everett with the Piraha

This is one of many cross-linguistic studies of how cognition is different between speakers of different languages. Another example is a comparison of English speakers and speakers of a language called Guugu Yimithirr (yes, real) which differs in its lexicalization of space. English speakers (e.g., me, possibly you) can use left and right with reference to the speaker itself (i.e., indexical). Guugu Yimithirr speakers use cardinal directions (North, South) even to speak about events on a small spatial scale.

This difference affects spatial cognition. Perhaps obviously, Guugu Yimithirr speakers are better at orienting themselves in open space since they must constantly be aware of where North is. Annecdotally, they are better than homing pigeons in orienting themselves to the compass absent any cues.

Conversely, English speakers are far better at a task that involves completing a maze and arranging shapes then turning around to face another direction and doing the same thing. Thus, the presence of indexical direction words (left, right) improves our ability to think about space indexically.

Language and Space
Stephen Levinson

Guugu Yimithirr kids pointing cardinally

Not only is the existence of a word important, but so too is how easily you can access it.

In another study, based on Daniel Oppenheimer's work on processing fluency, the researchers looked at short term stock price fluctuations and its relationship to the stock ticker symbol (e.g., MSFT for Microsoft, FB for facebook). What they found is that stocks with symbols that were easier to say did better imediately after IPO.

Predicting short-term stock fluctuations by using processing fluency
Adam L. Alter and Daniel M. Oppenheimer

And riffing off that general idea, I worked with some colleagues looking at the difference in hard-to-pronounce and easy-to-pronounce plural words (e.g., keys as easy versus busses as hard). What we found is that the plurality of objects for easy-to-pronounce plurals was remembered better (Learning to remember by learning to speak by Marc Ettlinger on Cog Blog)

Learning to Remember by Learning to Speak
Quora User, Jennifer Lanter and Craig VanPay

Children showing us what they know about one and many

One thing to note: most of what I'm talking about here is accuracy and you're talking about speed. In cog sci, the two are often interchangable because of a well-known speed-accuracy trade-off , Wickelgren77

What may be surprising is how quickly these effects begin to happen. Recent research has shown that when learning a new word (in your native language), the new word begins to affect cognitive processing as quickly as a day later. This is particularly true if you've had a chance to sleep on it.

Learning and Consolidation of Novel Spoken Words
Matthew H. Davis, Anna Maria Di Betta, Mark J. E. Macdonald, and M. Gareth Gaskell

The effects of sleep on the consolidation of novel words

Ultimately, your question serves as a useful companion to this other Quora question: Does an increased vocabulary change the way you think?

In answering, I suggested (Marc Ettlinger's answer) that there is some effect of an increase vocabulary, but "change the way you think" is a bit of an extreme way to put it; vocabulary doesn't serve as a straight jacket for what you can and cannot conceive of. Almost every invention was an idea before it was a word, for example.

But there is definitely an effect, as the answer to your question shows.
So, language can serve as a cognitive tool, improving your ability to think about the things language labels.