Store all entries in a single array from Wufoo api
Just some quick note:

1, I would separate the configuration settings to a distinct php file (and use a WufooConfig class/struct) and include it at the beginning of the code. It makes the configuration easier to change. If you use a WufooConfig class/struct and pass it to the wufoo_entries() function you will be able to use the same function to fetch data from more than one accounts. Furthermore, it would encapsulate configuration, so you just have to pass around one parameter instead of $account, $form, $api_key, etc.

class WufooConfig {
    private $account;
    private $form;
    private $api_key;
    ...

    public function __construct($account, ...) {
        $this-&gt;account = $account;
        ...
    }
}


2, Extract the then and else branches of the if in the wufoo_entries() function:

if ($refresh) {
    $results = fetchContent(...);
    saveToCache($results, ...); // write file here
    return $results
} else {
    return getCachedContent();
}


It makes your code easier to read, you'll see the big picture immediately, it won't be distracted with low-level API calls.

For the same reason I'd create some new functions, for example:

functio entryCount(...)
    $count = wufoo_api($api_key, $account, $form, $offset, $limit, $api_uri);
    $counted = $count['EntryCount'];
    return $counted;
}


The name of the function says what you want to get and the body shows how you do it.

3, Change 

for ($offset = 0; $offset &lt;= $counted - 1; ...


to

for ($offset = 0; $offset &lt; $counted;


It's more common.

4, In the inner for loop the $result loop variable is a little bit confusing, it's easy to mix up with $results. $resultIndex would be better.

for ($resultIndex = 0; $resultIndex &lt; $limit; $resultIndex++) {  
    array_push($all_results, $results['Entries'][$resultIndex]);
}


Finally, I agree with @Yannis Rizos, the whole logic would be better as a class.