The character didnt really make them just had a culture which at least in fantasy has far more availability of them, she was able to knock out much larger areas and use them in targeted and painful push manuevers in the show.
That's kind of nitpicking, isn't it? She didn't use arcane powers to make explosions either, but you seem to be ok with picking up artificer if it allows explosions, right?
I was just trying to answer the question that you appeared to ask of how you can mechanically achieve explosions for your character. The most obvious answer was, "let her make explosives"... and then she can use explosives. It's by no means the only option, but it's a solution that doesn't require taking the character into an arcane tangent.
You could also multiclass for some power effect that is similar. If you're tactical, then multiclassing into Wizard is reasonable and could allow you an encounter power of Thunderwave or Scorching burst (depending on if you wanted to go for burning them or knocking them back).
Going that route means you'd have to decide on which power you want, and can only swap between the 2 through retraining as you level, and you'll only have them once per encounter.
Going for Thunderwave means you'll need to boost your Wisdom a little, but sounds more useful for pushing (which you mentioned). Scorching doesn't do knockback, but it doesn't cost you additional abilities either.
Going for Alchemy means it costs gold, but you aren't limited to one per encounter, aren't limited to what type of effect you want your bomb to have (knock back, smoke, burning, etc).
Now if you just really want to stay true a specific concept that you want to mimic, and want to just obtain them without actually making them, then maybe your DM might let you buy them. However, your DM might make the deduction that by allowing you to obtain or buy as many bombs as you want might effectively circumvent getting a feat, and bring to question the value of getting a feat in the first place. If he does make this deduction, I'd be inclined to agree.