While it's potentially ambiguous, as Weather Vane said, to me it certainly seems more likely to mean "Fill out the entry form no earlier than 10 days before the flight"; I think it's fairly unlikely it was intended to express "at least 10 days before the flight".
As fev says, "within" pretty clearly means that there is some range of time in which you must fill out the form. The potential ambiguity is because in "within 10 days before the flight", the following noun phrase "10 days before the flight" has a form that would generally cause it to be interpreted as a point in time rather than a range. To determine which range of time is meant, it makes most sense to me to take the flight itself (or the date of the flight) as the other point, giving a range "from 10 days before the flight up to (the day of) the flight". The other interpretation would only be possible if you interpreted the other point as the current time ("from now up until 10 days before the flight") which doesn't seem natural to me.