Yes you can. I never go without bringing my own snacks and drinks
As you approach the front Gates of the Disneyland Park walking from the tram, you will see on your left as you approach the gates a sign that says picnic area. If you aren't looking for it you'll miss it but if you are it's not hard to find. Once you enter that area you will see picnic tables and large lockers big enough for a small ice chest. I have even talk to people who have sent one person in their party to go out and grab a bunch of pizzas and meet their group back at the picnic area. You can bring snacks and water make sure that everything is unopened. And don't bring anything you are prepared to be asked to dispose of before entering the park.
Yes
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