Paper Exams
I would strongly recommend that you find a way to ensure that students have paper exams. In my opinion, this is the best case for the vast majority of students.
If your institution / department cannot afford a printer / copier which reliably works, it might be well worth it to spend your own money to buy a cheap laserjet printer.[1] Alternatively, if you are in a large enough populated area, there should be some business which can make copies for you (Kinko's, OfficeMax, some local mom-and-pop, etc).[2]
You could save some of the printing costs by printing out only the questions without space to write answers, then have students write their answers on blank sheets of paper (which you can provide). This isn't my favorite approach, as I use the space on the exam to sort of hint about how long a typical answer should be, but not everyone does this, so it might be fine.
Or you could just put the exam questions on your LMS (visible only during the exam), tell students to access the LMS during the exam period (using whatever device they have access to), and not worry so much about cheating (by which I mean "do the normal things you would do to monitor your testing environment, but don't kill yourself).[3]
Putting Questions on the Board
As suggested by another answer, you could write the questions on the whiteboard / blackboard. Or type up the questions and project them onto a screen.
This is a viable option, though I think that it creates some problems with respect to accessibility, in that not everyone is going to have the same view of the board / screen. Moreover, if you are writing things on the board, your handwriting had better be immaculate.
Lockdown Browsers / Locked Down Computers
I have a great dislike for lockdown browsers. In order for a lockdown browser to do its job, it essentially needs to "root" the machine on which it is installed (the browser needs low-level access to the machine). This opens up all kinds of holes for other malware to get in, and I'm not even sure that I trust the browser publishers. These things just feel icky to me.
I avoid such tools as much as possible.[4]
On the other hand, if you are giving your exam in a computer lab, or if the machines being used are otherwise owned by your institution, it should be possible to place the exam on each machine (e.g. as a .pdf) and turn off all network access. You might talk to your IT department about how this could work (it is not an entirely trivial thing to do, but definitely possible).
Use Another Form of Assessment
I've had a lot of luck in the past with oral exams: I give students a list of five problems about a week before the exam date, and they have that time to prepare solutions. During finals week, each student makes a 20 minute appointment for the examination. At the start of the exam, a student may opt to:
remove one question from the list---I then roll a four sided die, and they answer whichever question corresponds to that roll; or
double up on a question from the list---I roll a six sided die, and they answer whichever question corresponds to that roll (on a six, they answer the question they doubled up on).
While they work through their solution, I ask lots of questions to see where they have difficulty. The downsides of this approach are that questions have to be well written (so that they potentially cover a large swathe of the material covered during a semester, and are open ended enough to invite questions during the oral examination) and it doesn't scale very well (I can't imagine giving this option to more than 100 students in a semester).
Have students end the semester with some kind of presentation. While I have not actually done this in the past, I think that it would be reasonable to give students a list of "difficult" problems / topics to choose from, then allow them to work in groups to solve those problems (or research that topic) and then present their findings to the rest of the class. In general, I think that it is not too hard to tell whether or not students understand the material well, and whether or not all of the students in a group of two or three have made real contributions.
Give some kind of take home "exam" where the questions might require some amount of research effort. If you know that your students are going to be using the internet, why not lean into it? Again, this requires a great deal of effort to craft the problems / research topics you are giving to your students, but this also has the potential to be a much more authentic form of assessment.
[1] I had a decent HP printer for about 15 years; when it died, I replaced it with a Brother printer---neither cost more than $300 new, and both could print off hundreds of pages fairly quickly.
[2] Contrary to one of the other answers to this question, I would not ask students to give you cash for this. Asking students for money leaves a bad taste in my mouth---while it may not be unethical, strictly speaking, it has serious feel-bads. Copies should not be so expensive as to justify those feel-bads.
[3] My experience is that students are bad at cheating. As an example, I teach a lot of asynchronous classes. All exams, other than the final, are taken in settings where I have no control at all over the testing environment---I assume that students are cheating like crazy. However, the ways in which they cheat are typically quite obvious---for example, a student giving me a decimal expansion out to 12 places has obviously used a calculator (which is typically against the rules); a student referencing kinematic equations in a Calc I class has obviously spent some time on Wikipedia (or similar). The students who feel the need to cheat typically lack the skills to cheat well.
Moreover, students only cheat because they think that (a) the assessment is so important that they must pass, and (b) they don't know the material well enough to do well. Lowering the stakes of individual assessments does a lot more to reduce cheating than anything else I've tried. To lower the stakes, try more, smaller assessments; more take home work; more work which can be done collaboratively; more opportunities for revision; and so on.
[4] I feel slightly different about the idea of a lockdown browser if the computers being used are owned and administered by your institution, but they still feel kind of icky to me. My preference is to find another way.