Customizing your cover letter and resume to the specific internship takes time and effort, and can help you stand out as a candidate. It shows that you have taken the time to research the organization and understand its needs. In your cover letter, focus on how the skills/experience/enthusiasm you have can help the organization meet its needs and mission. Be sure to express why contributing to this organization's mission is meaningful to you, and that you are excited to be a part of it. Companies want to have interns that are truly excited for the opportunity, and when you don't have a lot of work experience, your attitude and enthusiasm can go a long way!
If you aren't able to secure an internship, I would recommend that you pursue volunteer opportunities. Volunteering gives you the opportunity to be part of the organization and may offer leadership opportunities. You could help organize events, contribute to office management, or interact with the organization's employees and clients, and all of this can give you valuable experience and build skills that will make you more competitive for an internship or job in the future. Some volunteer activities can even be done online, and you can choose an organization or cause that you are proud to support!