I went to Japan and want to ask the food vendors whether I can eat in front of
his shop.  
Basically, I want to say "Can I eat here? or Is it ok to eat here?"  
I know that

> ここ = Here  
>  食べる = Eat  
>  大丈夫 = All Right / Okay

but I am not sure how to put them together?
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english-to-japanese
Can I eat here?/ Is it ok to eat here? in Japanese
You are almost there. Using your words, you could connect them as such.

> ここ **で** 食べ **ても** 大丈夫 **ですか** ？

Using も after after the -te form of a verb is useful for asking questions and
has a meaning similar to "even if".
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It can be also

> ここで食べてもいいですか？

or more casually

> ここで食べてもいい？
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Using all: ここで食べても大丈夫でしょうか？ ここで食事しても大丈夫ですか？

  * 食べて - te-form.
  * te-form and も - Even if ... , ...
  * I used ここ **で** , because there is no an adverb that means "here".
  * Its translation: Is it ok to eat here? (Is it ok even if I eat here?) 

* * *

In the following, I don't use them together.

  * ここで食べてもよろしいですか？
  * ここで食べてもいいでしょうか？
  * ここで食べていい？ (Friendly)
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With respect to the context of your question, if you would like to attach
importance to _where_ you are eating - in front of the shop / or in the
vicinity, you might want to modify your question to emphasize that it is the
_position_ (この場所, kono basho) you are querying about (and not necessarily the
act of eating, which you implicitly assume to be acceptable).

> この場所で食べても大丈夫ですか？

Otherwise, if you're worried about _eating_ any of the other answers are
acceptable.

I would say:

> ここで食べていい？

as you're eating at a food stall and it's an informal setting.

Nothing stopping you from being polite if you wish to be so.
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2017-01-18T15:55:29.723
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I'll add this as another answer. So far, all the other answers are addressing
how to ask "Is it **_OK_** to eat here?" However, as you know, Japanese people
are often indirect, so you might in fact say/hear this from the negative point
of view.

> ここで食べては **だめ/いけないん** ですか？ → Is it "bad" to eat here? (Should I not eat
> here?)

Taking it one step further, you might even assume it's not OK to eat here, and
state that fact as a semi-suggestive question to confirm your assumption.

> ここで食べては **だめ/いけないん** です **ね** ？ → I shouldn't eat here, right? (It's "bad"
> to eat here, right?)

* * *

If you take the direct route by asking "is it OK?", you may come across as
pushy or confrontational if they don't want you eating there. Even if it is OK
to eat there, you may still leave a bit of a bad taste in their mouth by
asking directly.

On the other hand, asking indirectly gives the impression that you'll politely
and willingly comply with their request if they don't want you eating there.

>   * You: ここで食べてはだめですね？ → I shouldn't eat here, right?
>   * 店長: あぁ、はい。すみませんが。。。 → Um, yes. I'm sorry... (please don't eat there).
>

And if it is OK to eat there, asking this way will make you seem more humble,
and they will likely gladly tell you to go ahead.

>   * You: ここで食べてはだめですね？
>   * 店長: いえいえいえ、全然大丈夫です。どうぞ！
>
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