Disclaimer: This is my first Malphite support guide so it might not be as "Rock Solid" as I claim. But, I love playing him as a tank support and thought I might share my joy with the world.
Why Malphite support?
1) His passive means you get a TON of armor; no need to be afraid of heavy hitter adcs like Caitlyn/Draven/Jhin. You can eat as much as those cupcakes as you want :)
2) He mainly goes in at level 6 and its a pretty strong all in and ganking support so mostly guaranteed kill
3) He offers pretty much nothing pre-6, and this is the beauty of it. Too many adcs i played with didn't know how to trade in lanes properly. And since everyone knows Malphite doesnt do anything pre-6,they have no choice but to farm safely thus preventing my adcs from dying on me too many times
4) Teamfights are awesome cuz everyone follows a Malphite for some reason. you may not get it with other champs like Leona. Also you are the main character since everyone is counting on you to start the fight.
5) If enemy is ad heavy and/or your team needs a tank, Malphite is pretty gold efficient due to his passive.

Q: Damages and Slows 1 enemy and gives you movement speed
W: Increases your armor passively, and increases your auto damage for a few seconds actively.
E: Damages and Slows attack speed of nearby enemies
R: You throw yourself to an area to damage and knock enemies up in the air. This is your main source of engage, and the start to your team's teamfighting strat. Usually other champions like yasuo and orianna will follow up after you ult.

Max R whenever possible. Usually I max W to get more armor to tank in enemy adc attacks. Plus this makes your armor stat more gold efficient. Then, max E 2nd if they have attack speed focused enemies (jax,xin,fiora, other adcs). Otherwise, maxing Q 2nd is good as well if you want to give chase since it gives a good slow.

Starting Items: Usually you want to go the relic shield for extra hp and its easier to proc gold income than spellthief (and saves you mana). If enemy lane seems weak, go refillable potion to save the environment. If they are poke heavy/all in heavy, 3 hp potions might be good as well.

You then build items depending on how the lane is going. Ideally you want a mixture of HP, ARMOR, and CDR (especially since you want to engage with your R as much as possible).

Then, get mercs if enemy is cc heavy/ap heavy.
I rush iceborn personally because it is a cool item and also because it gives 20% cdr, an aoe slow, armor , and the passive damage scales with your armor passive.
Then, it depends:
Are there AD enemies that heal a lot? Thornmail.
Are there AD enemies that crit a lot? Randuins.
Are there AP enemies that hurt a lot? Spirit Visage.
Are you having long fights and need sustain? Warmogs might be good.
Do you have AP carries in your team mostly? Abyssal mask is good.
Do you want to be like the AP Carry while being tanky? Liandry/Rod of Ages isn't too bad. Unfortunately, with your support gold Rod of Ages can't stack very well so it's better to buy the armor items first that can be bought in small parts as compared to a 850 g investment into the Blasting Wand.

Runes are mostly your choice. I prefer going Resolve and Precision. Guardian is good because it gives your carry a shield (since you can't help them in lane otherwise). Demolish is good to demolish towers. Conditioning is good for getting more resistances and helps bring your armor up. Revitalize is good for regen and shielding.

Tenacity is good since you can just focus on getting ninja tabis and not worry about mercs. Triump is good to get extra gold, but mostly to keep you in the fight longer. As a support, you have low gold access meaning less defences and staying in fight longer will be good for you.

You can go Inspiration secondary if you want extra potions for lane sustain. Since Patch 9.23 removed heals from relic shield, this might be a better option sometimes.

Are you scared of AD champions because they hurt too much and as a support, you don't have gold to buy armor and you just want to stay alive? HAVE NO FEAR, SUPPORT MALPHITE IS HERE!!!
Have fun playing him and feel free to experiment. Feel free to leave comments below and if this gets received well, I can write more guides for other champions I play. And if this doesn't go well? Then, I retire from writing guides. Maybe. We'll see.
Thanks for reading!