I am basing my answer on your profile, which says you're studying computer science in Germany. Obviously, the answer will vary depending on your program and location. Here we go:
The semester ends right before Christmas and after it I will have time
until April as the semester in my home country does not start earlier.
So I am considering to do an internship in the U.S.
That's a pretty short timeframe to do an international internship. Many companies already hesitate about hiring internationally when there are local students who can also do the job, especially given the costs involved (unless you want to pay for your travel expenses?). As a result they may expect a longer work term from you to make it worth their while. Put it this way. If you were the company and want to hire an intern for a few months, would you go through the trouble of bringing someone in from thousands of miles away when you could just hire locally? In general you will need to prove exceptional qualifications and often a longer term commitment than a few months to have a shot as an international student. If not, you did mention that you are eligible to extend the duration of your stay. This might be pretty important. Also keep in mind that many smaller companies with limited funds might not be willing to hire internationally at all, so you may be restricted to larger ones. In any case, you absolutely have to make it clear to any companies you apply to where you are coming from and what timeframe you are available. This should be clear on the resume, so there are no surprises.
If I misunderstood your post and you are already in the US though, you can ignore the above point.
Also, if you are planning to apply to start after THIS Christmas, you might already be too late. I have applied to US internships from Canada, and even here they often get their hiring done by early October for hires starting in January. I imagine international hires from outside the continent require even more paperwork and processing time, even if you already have your work visa figured out (unless you're currently residing in the US). But the only way you will know for sure is to contact recruiters at your chosen companies and ask them what their policy is on this. They may be able to find a solution for you.
I should add this: have you considered looking at internship opportunities in Germany? There are many large US companies with offices in Germany, and this may be a good way to get your foot in the door so you can later move on to a US office (if you prefer this). It should be significantly easier to get a short-term job in Germany than in the US if you are studying there. But I don't know how internships work in Germany, so perhaps this isn't an option.
Are there certain web portals you can have a look at to find a job or
is it better to directly seek for an employment on the companies's
Many job postings do indeed get posted online, and many companies offer a way to apply through their website. But getting an interview this way is a bit like winning the lottery. Large companies get a huge volume of applications and many of them get filtered by some algorithms which check for certain key words, so the recruiters might only see a small sample of the resumes submitted. In general, the best way to get hired is to actually network with people at the company. Since you presumably can't just buy a plane ticket to the US to go to some job fairs, you can try networking with recruiters at companies with offices in Germany and see if they can put you in contact with their counterparts in the US. Better yet, if you go to a reputable university (or know one nearby), you may already have recruiters from these companies visiting the campus. If this is the case, find them and talk to them, describing your qualifications and why you're interested in the position, and submit your resume/CV (keep in mind US resumes are very different from German CVs).
If you can't find anybody in Germany to network with from the company, you can try emailing recruiters. Don't just submit your resume though, especially if there is a more "official" application process that they might want you to follow, but take the opportunity to get your concerns clarified with them about getting an internship there by asking some questions (keep it short and to the point!), and attach your resume for reference if they are interested in continuing the process. Again, be sure to describe your interest for the position and qualifications. Keep in mind that many recruiters get hundreds of emails per day though, and it's still quite possible for your email to get lost.
My biggest disadvantage definitely is that I am not a native speaker.
I believe I can understand and speak English very well. However when
talking to me you quickly get that English is not my first language.
Can this be a big disadvantage?
Based on the post you've written, you should be okay. A lot of people join large US companies from around the world and you can be sure that they have accents. As long as you can communicate with people, nobody will mind that English is not your first language. It could even be an asset if they need someone who speaks a foreign language on their team.