Depends on how big a stickler for timing you are, or how willing you are to abuse convoluted item holding rules to your advantage.
Here is a rule clarification regarding when an improvised weapon counts as a weapon:
Now in order to cast Green-Flame Blade or Booming Blade, you need to have a material component or a focus (which in this case can be a weapon). However, an improvised weapon only counts as a weapon while you are attacking with it. However, however, whenever you cast either blade spell: "As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails."
Some DMs will let you get away with it because an action is basically an instantaneous effect, and all the triggers would therefore happen simultaneously. Other DMs would say "No, you need to have the components available before you start casting the spell." Meaning you would need both a normal weapon (or focus) and an improvised one to trigger the spell, and also an improvised weapon and a free hand to start the grapple. The trick with that set up is that you can drop an item as a free action between the Action to cast a spell, and the Bonus Action granted by attacking with an improvised weapon.
Assuming that is too much juggling for your blood, or you just don't want to drop anything, there are two ways to get around that.
Firstly, you can offset your requirement to hold a focus in various ways. Some classes (Like Clerics and Artificers) can wear a focus. Sorcerers can use Spell Points. And there are a few items that can functionally do the similar tasks.
Secondly, you could use natural weapons. They always count as weapons. Allowing you to prep the spell with your claw, make the attack with a bottle(or whatever) then make a grapple with your empty hand (which can also be the same hand that had claws on it)
