Yes, you can take part in Counselling. In online Counselling you are not asked to verify or submit any documents so there shall be no problem. Even many will not have their board marks by then. Once Counselling is done, you will be asked to report at a paurticular place say the college you chose. There you need to produce all the original documents. By then, your board markNEET should be ready else the authorities have full right to reject your admission.