Both are acceptable depending on the meaning.
With won't (will not), the meaning is that you will not need to go out to get some food, because there is already food.
With don't (do not), the meaning is that you do not need to go out at all, for food or any other reason, because there is food.
ell.stackexchange.com/users/38266/NibblyPig
I think the first sentence is correct, since you used "will" in the first part of the sentence it would be more appropriate to use will in the second part
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Neither is good to my ear.
You will find some food so that you won't/don't need to go out when you arrive.
"
So that " is used to indicate a purpose or to give explanation .
Neither of these definitions are applicable here.
Instead of " so that " you need to use " so " which shows the result (therefore) .
You will find some food so you won't/don't need to go out when you arrive.
In this case " won't " will be better since we are talking about a future time " when you arrive ".
" don't " will be better about a general statement or a statement applicable in the present:
I payed the bill morning
so you don't need to worry about it.
I will hide the sweets so that you don't overeat them.
P.s.
You might try " You will find that I've stuffed plenty of food in the fridge so you won't need to go out to buy it when you arrive. "
ell.stackexchange.com/users/44134/SovereignSun
I think it is better grammar to use future + future , rather than future + present .
You may use future + present in some situations for some specific meaning, but the broad rule is future + future .
However, I would drop that without blinking - it sounds strange.
Use so instead of
so that in this case.
Extra:
so / so that do(es) not require anything.
It is the verb in the main sentence which requires something.
See same example, past tense:
We found some food so we didn't need to go out when we arrived More: will is future, when you arrive is present, but suggests future, therefore it is natural to use future for the middle verb too.
ell.stackexchange.com/users/89027/virolino
When the time of a subordinate clause is the same as of the main clause, the verb form in the subordinate clause is usually simplified.
So Michal Swan says in his Practical English Usage (§580.1).
In the OP's example, however, the subordinate clause is talking about further into future than the main clause, so it's easier to digest when it says won't .
ell.stackexchange.com/users/6384/karlalou
You will find some food so that you  won't  need to go out when you arrive       or   You will find some food so that you  don't  need to go out when you arrive.   Which one is the best? I think first one is better.
Do I use present or future with "so that"?
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<tense><future-tense>
ell.stackexchange.com/questions/100825
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With won't (will not), the meaning is that you will not need to go out to get some food, because there is already food.  With don't (do not), the meaning is that you do not need to go out at all, for food or any other reason, because there is food. You may use  future + present  in some situations for some specific meaning, but the broad rule is  future + future .  Instead of " so that " you need to use " so " which shows the result  (therefore) .   With " so that " you need a purpose (outcome and reason):   He locked us inside  so that  we wouldn't go to the party.  You might try " You will find that I've stuffed plenty of food in the fridge so you won't need to go out to buy it when you arrive. " While both are acceptable, "won’t" means that the person will not have to go out for food, whereas "don’t" means they do not have to go out for anything. In terms of grammatical rules, when a subordinate clause and a main clause share the same time, the verb in the subordinate clause is usually simplified. That said, in this case it is perhaps better grammar to use future + future instead of future + present, because the latter is reserved for more specific meaning. A solution is to substitute "so that", which needs an outcome and reason, with "so", which indicates a result, because a native speaker could argue that neither option sounds natural.
While both are acceptable, "won’t" means that the person will not have to go out for food, whereas "don’t" means they do not have to go out for anything. When a subordinate clause and a main clause share the same time, the verb in the subordinate clause is usually simplified. That said, in this case it is perhaps better grammar to use future + future instead of future + present, because the latter is reserved for more specific meaning. A solution is to substitute "so that", which needs an outcome and reason, with "so", which indicates a result. Neither option sounds natural.