The Blinding Mage will have the opportunity to tap the Furyblade Vampire after its controller has decided if they will discard a card or not.
In Magic who gets/has to act is based on a system called priority. The super short version of priority is a player needs to have priority in order to cast spells or activate abilities. Before any spell or ability resolves all players must pass priority without doing anything. If priority is passed by all players without anything being on the stack the game moves to the next step or phase.
The Combat Phase has several steps inside of it, the first two the Beginning of Combat Step and Declare Attackers Step are the ones that are relevant to this question.
The Vampire's ability triggers 'At the beginning of combat on your turn', this means that it triggers during the Beginning of Combat Step. After the ability resolves (and you have chosen either to discard a card or not) your opponent will receive priority (they technically also get priority before the ability resolves, but since they wouldn't want to use their Mage's ability then I'm skipping over it), which means they have the opportunity to cast spells or activate abilities like Blinding Mage's.
The Declare Attackers Step is when you would actually declare the Vampire as an attacker if it is not tapped. Note that you can't just try and do something like 'I discard a card to the Furyblade's trigger and attack with it.' to try and skip your opponents ability to tap it. If you try and do something like this they have every right to back the game up and tap your Vampire.