I have a 2nd level wizard in D&D 5th ed. Looking through the spell lists I can't find any healing spells that a wizard is allowed. There are healing spells in the full list but they are not in the allowed list for a wizard. Am I missing something?
No, you're not missing anything. Wizards never had healing abilities in this edition*.
Lacking any more concrete designer reason for this, here's a text from the DMG (page 283)
... Wizards and Sorcerers don't typically have access to healing spells, for example, and adding a healing spell to the wizard class list would step on the cleric's turf.
So, if you want to use healing spells, go for any of the classes with those in their spell list: Bard, Druid, Ranger, Paladin, Cleric. If you want to play a spellcaster that can also heal, there are a number of options available to you, you could multiclass into the above until you get some healing spells(though that would make you MAD- Multiple Ability Dependent), you could play a Divine Soul Sorcerer, and there's also a Warlock archetype in Xanathar's Guide to Everything.
I don't have the books on me right now, but - though not explicitly a pure healing spell, quite the contrary -, Vampiric Touch does allow a wizard to regain HP siphoned from another being not immune to necrotic damage (necromancer).
You can also use a Wish to produce healing effects, in truly dire straits. :)
As another edge case, you might consider Polymorph too, as it provides a new HP value to the subject for the duration of their transformation, and damage gets carried back over to the original form only if the new form is lost due to losing all its HP, and even then only the excess damage is applied to the original form.
While none of these are healing spells in a strict reading, they all allow the wizard to apply a healing effect to their subject.
As others have said, wizards do not get any kind of classical healing spells. Their support role is traditionally in the form of utility and battlefield control.
But you can multiclass to get those
However, should you want those spells and should your DM allow it, a single multiclass level into bard or cleric will give you access to Cure Wounds and Healing Word. Multiclassing into those classes also doesn't reduce the number of spell slots you have access to and give you some additional abilities.
Both classes give you 2 known level 1 spells. Bard gives you Inspiration dice and one additional skill proficiency. Cleric gives the chosen domain's spells as always-prepared spells. In the case of the Life domain (which would probably be your focus), you would most notably gain Cure Wounds, and +3HP on your Cure Wounds spell if I understand the Life feature correctly.
I must repeat that multiclassing is an optional rule left to the DM's discretion. But if multiclassing is allowed, a very low level investment allows a wizard to cast healing spells without compromising his number of spell slots.
Or take the Magic Initiate feat
As noted in the comments, the Magic Initiate feat gives you access to 2 cantrips and a level 1 spell from the class of your choice. As a caster, the spell is added to your known spell list and you can cast it once per day for free.
Feats are also an optional rule, so check with your DM if they're allowed.
This is a design decision intended to keep the roles of the different classes distinct. From the Dungeon Master's Guide, in the section Creating a Spell:
Make sure the spell fits with the identity of the class. Wizards and sorcerers don’t typically have access to healing spells, for example, and adding a healing spell to the wizard class list would step on the cleric’s turf.
Of course, the Divine Soul Sorcerer from Xanathar's Guide does exactly this, thus proving that guidelines are guidelines, not laws which a DM can't alter — but note that giving the sorcerer healing powers changes the core identity of the class. Any change for a wizard should probably be on similar scope, requiring at least a new Arcane Tradition and trade-offs against other class features.
If your GM allows Unearthed Arcana material there is Theurgy Arcane Tradition, if he doesn't you could drop the wizard and take the sorcerer class and choose the Divine Soul Sorcerous Origin, both allow to cast a certain number of cleric spells as arcane spells. Sorcerer class has pretty much the same spell list as the wizard.
One of the Unearthed Arcana suggested "Healing Elixir", a 1st level spell that would create a minor healing potion that would last for 24 hours before vanishing, available to wizards and warlocks. It's not legal for Adventurers League or otherwise officially sanctioned, but your DM might be persuaded to include it. (I allowed it in my campaign, where the wizard was the only caster -- the cleric's turf is thus safe)
It's an interesting choice because as far as I can tell it's worse than all other healing spells. It has a 1 minute cast time (therefore non-combat), it doesn't scale with spell levels, it heals 4-10 points, which is generally worse than the 1d8+stat points from cure wounds and definitely worse than the straight 10 from Goodberry.
Could a wizard healing spell exist, yea, but does it exist in the current 5e system, no. It could exist, but really its up to the DM if there was a need for the story or the group needs more healing. So, ask the DM, give a good reason, you don't have to slave to the rules. But, "I just want a healing spell", is not a really good argument.