I was planning to trade LTC/USDT pair and discovered that Tether.to is not accepting new registrations. Is there any way I can use it? Can I still get an online wallet like omni wallet or exchanges wallet and store it there? Does omni or exchanges give private key to the wallet? I'm a complete noob here, enlighten me please

You can still use it, but only between exchanges.
When you open an account in some of the USDT supporting exchanges (list here:
https://coinmarketcap.com/currencies/tether/#markets), when you click on depositing Tether, they will generate you an address.