Is it possible to write a literary analysis on a sonnet without a thesis statement, just to analyze the text instead?If so, how would I do so?
You should always have a thesis statement, regardless of whether or not you state that thesis openly. A thesis statement can be hidden, implied, or unstated. Most teachers prefer to have the thesis stated openly, of course.
The value of a thesis statement is twofold. For one, it orients the reader; the reader will know what to expect, even if all of the details aren't spelled out entirely (they shouldn't be!) in the thesis statement. The thesis statement also helps the writer stay focused. It's easy to get lost in one tangential thought after another; a strong thesis statement (whether written at the start or much later in the writing process) helps the writer stay on track. I know that I tend to write a strong thesis statement pretty late in the writing process. I also tend to put my thesis statement at the end of one or two engaging paragraphs. I try to get the reader interested and oriented before I spell out what I'm going to do in the entire essay.
"Just to analyze the text" is a pretty unfocused goal. Analysis can mean all sorts of things. At the very least, you could have a broad thesis statement that says something to this effect: This sonnet can be discussed meaningfully in terms of both its form and its content. (As a reader of poetry and analyses of poetry, I really enjoy essays that are able to make meaningful connections between the form and the content of poems.)