I'm guessing you/your wife provided immigration sponsorship. If so, one thing you/her/both had to sign was an "affidavit of support" that says that you will provide her basic needs and she won't be a burden on the taxpayer. This holds as long as your mother in law is not a US citizen and doesn't have 40 Social Security credits.
So in light of that, let's read your questions again:
Does Obamacare mandate that she get insurance?
She is my dependent, but am I supposed to provide insurance for her?
As said in the affidavit of support, if she cannot provide to herself then whoever signed the affidavit is to provide for her. However, some States do provide aid despite that, so check your State/local welfare rules.
She is senior. Can she get Medicare, despite being in this country for so little a time?
Only after 5 years as a Permanent Resident. Again, some States sponsor Medicare/Medicaid for seniors even if they don't qualify by the Federal rules.
Can I keep her without insurance?
Of course. But what happens if she gets sick, as older people occasionally have been known to get? If she ends up neglected in an emergency room, you may get into a nasty situation defending yourself from the criminal elder abuse charges.