There are many aspects to this request that you need to be aware of.
For example, what sort of screens is everyone else using? If people who've been there for 5+ years are also using 19" screens, and you come along asking for larger ones, then you're practically guaranteed not to get them.
If, however, you're the only one who has small screens, then perhaps they will buy you new ones. Remember, however, that you're not even past your probation, so your request isn't likely going to carry much weight.
Furthermore, the fact that they're not bothering to buy professional software should be a big warning sign to you that these people are not prioritizing your needs or wants as a developer. (if this is because they just don't understand the difference yet, they you should explain it to them ASAP)
At the end of the day, if you do decide to request new screens, be very polite and respectful. Ask to meet with your manager/supervisor, and express your opinion. If you can find some sort of study recommending screens of a certain size for development purposes (is there such a thing?) then you can maybe show it to management at that time.
An aside based on personal experience:
I've worked a co-op in circumstances very similar to what you're describing. I was the only developer in a small company where I was expected to program in the free edition of Visual Basic, and my desk was a folding table.
My advice to you is to get some decent experience (finish a project or two for them), and then bounce as soon as you can (after a year or two at the most).
In this sort of environment you won't have anyone else mentoring, or guiding your growth as a programmer, and it's very easy to stagnate, or pick up bad habits. These people are probably not used to working with developers, and will not understand why you might need to take your time setting up the foundation for a piece of software, taking your time to design something before implementing it, etc. (and no one will be around to teach you how to do it properly either)
These sort of places are usually characterized by a constant pressure to get something on screen, in order to demonstrate progress to managers who don't know the first thing about development, and think it's synonymous with IT support.