Yes, it's perfectly possible. Many of the well known internet brokers offer internet accounts for ltd companies
The downside is that a ltd company has no personal allowances for CGT or dividends. So they will pay corporation tax on the entire profit. And then you'll pay more tax when you take the money out of the ltd shell. Individuals have personal allowances (and ISAs and SIPPs) which means shareholdings are charged far less tax for individuals than for companies.
Also - a company which does no trade
other than investing its own cash in shares would be a "close investment company" (google it) which is subject to
even more tax
So as bluedonkey hints, there are good reasons not to.
It
might be a better idea to take the money out and invest it personally. Though there are exceptions - such as if you are currently a high/very high level taxpayer and want to leave some money in the ltd company for later.
Gryff