I want to play Blitzcrank in a support role. Should I start out with a Philosopher's Stone like other supports, or should I get a Tear of the Goddess for the mana to start off? What are other core items that would make Blitzcrank a viable support, and why?
As Drmixable mentioned, it is very situational. If you're intending to solely be a kill lane, it might make sense to go Sapphire Crystal, ward, health pot, then tears.If you do this, you are gambling completely on your lane getting first blood and then compounding that advantage into laning dominance for the rest of the laning phase.
If you are wanting to be a more traditional support early game, then definitely do the standard Faerie Charm, wards, health pots mix.
Even if you do start with the mana crystal and go straight into tears, I think it still makes sense to move towards the more traditional support build after you pick up the tears. Philo stone, Heart of Gold, lots and lots of wards, oracles, etc. Build your gp10 items into their upgrades, prioritizing Shurelya's if your team is winning team fights and Locket if they are not.
And - as always - buy boots on your second trip back to base. For Blitzcrank, Boots of Mobility upgrade later makes a lot of sense for getting into position to yoink someone or knock them up to protect your carry. However, if the team is full of hard CC you will of course want the tenacity from merc treads instead.
Other good items include getting an early catalyst(which will build into Banshee's later), to sustain your lane harass and increase your passive shield proc if necessary. An aegis, of course, is always welcome from a support. Frozen Heart could be viable if the enemy team has more than two sources of straight auto attack damage, but generally won't be worth it.
Hopefully this wall of text gets you started on the path to thinking about how to apply your items to each situation. There is no single item guide that any hero should follow. Anything from who you are facing, how much gold you have, and how late it is in the game can vary your items and the order you build them in.
It as all situational and a build you find online is never fool proof. I don't play blitz much but here are a couple guides I found