Yes.

Having a natively-developed orthography (writing system) is relatively rare, and while a majority of major languages have borrowed or inherited orthographies from other languages, there are still very many that lack either a standard orthography, or any standard at all. One of my many hats is as a documentary linguist, working on undocumented or under-documented languages, usually small and spoken in far away areas, and the vast majority of these lack a writing system. One of the tasks that's being undertaken by both missionaries and community members is to develop standard orthographies and try to get community support, but this is a huge task and without some governing body to enforce it, it's difficult for these to be successful. I've spent a good chunk of the last 3 months pulling my hair out about this.

So how many languages is this? Most estimates put the total number of languages today at around 7000. There are issues with this number but let's just say it's fine for our purposes here. Of those 7000, around 10% lack any sort of writing system, 50% have documented writing systems, and for the remainder we have no data. Think about that a second. 40% of the known languages have no documented evidence showing whether or not there's a writing system. This is based on Ethnologue's calculations, which again can be problematic, but it's not far off. A good number of those of course have non-standardised writing systems using whatever alphabet the dominant language in those areas uses, but not anything that's widespread or consistent.

Beyond that, a lot of languages in the past had no writing system until one was developed. The majority of Germanic languages had no writing system until well after the time of Christ, and these developed primarily through contact with Latin.

As for languages which are written but not spoken, this is also not such an uncommon thing. You can address this a couple different ways. There are languages which are written and spoken by one group but written and not spoken by another. This is the case with the role of Mandarin as the written standard in China, where many people who don't speak it fluently / natively / at all can read and write it. Before this, Classical Chinese served as the lingua franca for most of East Asia from Japan and Korea to Vietnam and beyond, despite not being something people were otherwise speaking to each other. It has been for the majority if not entirety of its history a written-only language, recitations aside.

A great many liturgical languages also fit into this description. Before the late 19th century Hebrew was primarily written-only, having previously fallen out of use. Of course it's spoken natively now, but this is a more recent development.

This brings up a difficulty in addressing your question. Namely, if Hebrew is what we call a written language and Hebrew is also what we call a different but similar spoken and written language, how do you determine where to draw the line? Sanskrit existed as a highly complex liturgical language without much in the way of speakers. That is, there were speakers, but what they were speaking differed in many ways from the written form. Certainly the roughly 15000 native speakers of Sanskrit today aren't really speaking Sanskrit in the classical sense.

Anyway, tl;dr: Yes.

Hope that answers your question.

Also, the following isn't to answer your question since it's just an oral tradition but it's a neat one that relates, so I'm gonna share it anyway since I've already typed it up: There's an history to explain the lack of writing, which is common among various groups in Northeast India / Northwest Burma / Southwest China. Most recently I've heard it told by Tangsa speakers and Adi speakers. It tells of a time in the past that God (or whoever) was handing out writing systems and while most were given their scripts on wood or stone, this group (whoever happens to be telling the story; it's not limited to one language group) arrived late so all there was left was animals skins. Then, in a time of great famine when there was nothing left to eat they were forced to boil the skins to survive. To put it another way, they ate their words. This is the sort of folk explanation that's become somewhat widespread, and I've heard it told by a number of people in unrelated language/culture groups.

Also, in case you saw it from another response that's since been removed — Another commenter pointed out that computer programming languages count as written-only languages. This is not accurate. Computer programming languages are not languages in the same sense that French is a language.

Someone also said signed languages would count as not spoken, but this is also not really how this is discussed since signed languages are effectively still spoken languages, just ones spoken in a different modality. There's been a lot of good research, some of it very recent, on this topic, and within linguistics a signed language and a voiced language both count as "spoken".