What is a word that describes a person who hurts people without meaning to (as in not intentionally)?
"Reckless", "careless" or "negligent" all mean roughly the same thing:
"Not giving sufficient attention or thought, especially concerning the avoidance of harm or mistakes." (Wiktionary)
The reckless person might hurt themselves or other people - that's ambiguous.
You could you "clumsy" for hurting physically, and "tactless" for hurting emotionally.
See the following from subscription-only LDOCE:
tactless (adj.): someone who is tactless says or does things that offend other people or hurt their feelings, without intending to and without realizing that they have done it:
He made all sorts of tactless remarks about her appearance.
-> it's tactless (of somebody) to do something:
It was a bit tactless of you to start talking about her ex-boyfriend.
moving or doing things in a careless way, especially so that you drop things, knock into things, etc.:
A clumsy waiter spilled wine all over her new skirt.
a clumsy object is not easy to use and is often large and heavy
a clumsy action or statement is said or done carelessly or badly, and likely to upset someone:
David made a clumsy attempt to comfort us.
P.S. "insensitive" is another related word, however it doesn't exactly imply that the person is actually doing something hurtful, but just not noticing how the others are feeling hurt.
One who unwittingly hurts people while going about their daily business can sometimes be called "train wreck", a "ticking time bomb", etc.
Oblivious, I believe that is the word you might be searching for.
oblivious |əˈblivēəs| adjective not aware of or not concerned about what is happening around one: she became absorbed, oblivious to the passage of time | the women were oblivious of his presence.
oblivious adjective oblivious to the conversation around the table | utterly and happily oblivious of the effect he was having upon his audience: unaware of, unconscious of, heedless of, unmindful of, insensible of/to, unheeding of, ignorant of, incognizant of, blind to, deaf to, unsuspecting of, unobservant of; insensitive to, unconcerned with, impervious to, unaffected by, indifferent to; informal clueless. ANTONYMS aware, conscious, sensitive.