I like the spell produce flame for flavor reasons, but it seems to be available only to Druids. Is there a way for an arcane caster (particularly a wizard or sorcerer) to acquire it in such a way that they could cast it using their intelligence/charisma? If I understand correctly, if I multiclassed as a druid, I could pick up the cantrip, but it would be cast with Wisdom, which isn't a desirable option in this case.
A Warlock with Pact of the Tome can add Produce Flame to their Book of Shadows as a Warlock cantrip. I'm not aware of another way to get the spell to be on your class list without being a Druid or Nature Cleric (who also can learn Druid cantrips).
Produce Flame is basically a weaker version of Light combined with a weaker version of Firebolt. You could easily style your light spell to be flames emanating from a ring your character wears, for example, which would produce a similar, but stronger effect.
As far as I know, there is no way to cast Produce Flame as a Wizard spell, i.e. using your Intelligence Modifier.
There are a lot of ways for a Wizard to learn Produce Flame, and some of them let you use Charisma as the spellcasting modifier instead of Wisdom.
Multiclass as a Druid
Self-explanatory. One level of Druid will let you choose some Druid cantrips, which will include Produce Flame if you so choose.
Multiclass as a Bard
This is a bit more difficult since you'll need to invest at least 10 levels (or 6 as a Lore Bard) but you'll be able to learn a Charisma-based version of Produce Flame.
Multiclass as a Warlock
Warlocks can also learn a Charisma-based version of Produce Flame by taking the Pact of the Tome as their level 3 option, and adding Produce Flame as one of their cantrips gained through this feature. Easier than the Bard option in terms of raw levels spent.
Replace an Ability Score Increase with the Magic Initiate feat
The Magic Initiate feat can be taken to gain access to Druid Cantrips/a single 1st level spell, which is like multiclassing into Druid but without sacrificing any valuable Wizard features.
Replace an Ability Score Increase with the Spell Sniper feat
Like the Magic Initiate feat. The advantage to this version is that it also doubles the range of all your attack-roll based spells. The disadvantage is that it only gives you the one cantrip of your choice (Produce Flame, in this case) and doesn't give you any other cantrips or spells, like you'd get from other options.
Multiclass as Lore Bard
You can multiclass into bard and pick the College of Lore subclass. This is an extreme method, as it requires you to put 6 levels into bard, but it does give you the Additional Magical Secrets feature:
At 6th level, you learn two spells of your choice from any class. A spell you choose must be of a level you can cast, as shown on the Bard table, or a cantrip. The chosen spells count as bard spells for you but don’t count against the number of bard spells you know.
This will allow you to pick up the produce flame cantrip and cast using your Charisma instead of the Druid's Wisdom.
Fire Genasi, a race found in the Elemental Evil Player's Companion, have the produce flame cantrip as a racial trait. This would use your Constitution rather than Intelligence or Charisma, so it doesn't match your parameters precisely.
Fire Genasi make good wizards with their +1 Intelligence, and +2 Constitution will help your Produce Flame be as effective as possible.
As with most things, talk to your DM
Others have covered RAW methods (e.g. multiclass into lore bard or warlock), but if this really is for flavor and your game has heavily story-driven RP like mine, then the best option is to talk to your DM.
My DM allowed me to teach our Druid the spell light because it made sense narratively that I would have tried to teach her a spell. Justification being that Cantrips are meant to be either simple or well-known spells that can be performed as training exercises or without thought.