A corporate that has an ISDA master agreement to trade Interest Rate Rwaps (IRSs) with a bank will undoubtedly be capable of also trading Overnight Indexed Swaps (OISs), as will any type of counterparty for that matter.
A corporate whose loan is tied to floating LIBOR will hedge using an IRS to convert to fixed. Hedging with an OIS would introduce unnecessary LIBOR / OIS basis risk.
The OIS market is for overnight unsecured lending. How the OIS fix is determined varies in different currencies.
Generally OIS is lower than LIBOR. However this is not always true. 1M EURIBOR was lower than EONIA (EUR OIS) for a prelonged period of time, primarily for two reasons; the inherent differences by which EONIA and EURIBOR fixings were calculated, and the LCR (Basel III Liquidity Coverage Ratio) which creates a natural aversion to borrowing money for a term of only 1M, hence the demand is restricted and the rate fell.