If I buy a $200 call option for $22 ($0.22) that expires in 3 weeks, with a current share price of $140, can I sell the option two weeks from now if the share price has reached $180?
The option you buy for $0.22 will steadily decrease in price towards zero while the share price is below $200. If the stock price goes up, the option price may go up a small amount in line with the stock price movements (this is called delta).
While there are market makers quoting bids and asks on the market, they are only required to quote $5.00 wide. If the price is below $5.00 then the market maker only needs to provide offer to sell. There is no guarantee that there will be anyone else in the market willing to buy it at that price. In practice the spreads (difference between the bids and asks) may be narrower so you could close the position, but there is no guarantee.
I'm asking because if it reaches $200, I will not have the money to buy 100 shares at $200 even though I believe in the company.
While the market maker is not obliged to bid at that price, you might be able to sell it if it is in the money.
However, if the stock closes on expiration day above the strike price of the call option, the OCC will automatically exercise the in the money options, which @Bob_Baerker mentions will not work out well.
This can be avoided by giving "contrary exercise advise" to your broker which results in an instruction the OCC not to exercise your option, if you are unable to sell the option on the market.
However, if the strike price on expiration day is above $205, there should be a market maker bidding at least a penny for it (if not, complain to your broker, who can complain to the exchange, who can tell a market maker to start quoting that series). Market makers are required to quote at least $5.00 wide unless it is a high priced security.
You should to go through the procedures with your broker before opening the position because the deadlines are strict (usually 4:30 at the latest) and they are likely to be busy during that time.
While technically possible to execute, the trade seems a risky one to do in that you are relying on there being sufficient liquidity to close out the position before execution, and due to the lack of liquidity may be required to issue contrary exercise advice even if the option is in-the-money.