Red blood cell on the left, white blood cell on the right, platelet in the middle. Wikimedia

Anecdotally:
A 65-year-old man died from blood clots following his second Moderna COVID vaccine. The clots were linked to the vaccine:

Thrombosis With Thrombocytopenia After the Messenger RNA–1273 Vaccine, Annals of Internal Medicine, 29 June 2021

There have been thousands of reports of blood disorders in CDC’s adverse events reporting system (VAERS) for all three vaccines: Moderna, Pfizer, Johnson and Johnson. From an article dated 29 June:

A search of VAERS, using search criteria including reports of blood clots associated with blood coagulation disorders, produced a total of 6,787 reports for all three vaccines from Dec. 14, 2020 through June 18, 2021.

Of the 6,787 cases reported, there were 2,893 reports attributed to Pfizer, 2,394 reports with Moderna and 1,459 reports with J&J.

Clotting was suspected early on:

I am a pediatric specialist caring for children with the multisystem inflammatory syndrome (MIS-C). I am concerned about the possibility that the new vaccines aimed at creating immunity against the SARS-CoV-2 spike protein (including the mRNA vaccines of Moderna and Pfizer) have the potential to cause microvascular injury to the brain, heart, liver, and kidneys in a way that does not currently appear to be assessed in safety trials of these potential drugs.
– FDA, Comment from J. Patrick Whelan MD PhD, 20 December 2020

Thousands of clotting cases later, it’s still suspected. The FDA hasn’t reponded:

The U.S. Food and Drug Administration (FDA) has not responded to concerns that mRNA products, through spike proteins, have “the potential to cause microvascular injury [inflammation and small blood clots called microthrombi] to the brain, heart, liver and kidneys in ways that were not assessed in the safety trials.”
– Blood Clotting Needs to Be Watched with All COVID Vaccines, States the Association of American Physicians and Surgeons (AAPS), 5 April 2021

These clots can appear anywhere: brain, heart, liver, kidney. What’s causing them?

CoViD vaccines may also directly infect platelets and megakaryocytes triggering mRNA translation and consequent spike protein synthesis intracellularly. This may lead to autoimmune response against platelets and megakaryocytes resulting in reticuloendothelial phagocytosis and direct CD8+ T cell lysis.

CoViD genetic vaccines may have a direct role in spurring autoimmune response against platelets that may clinically manifest in thrombocytopenia [very low platelets], haemorrhage [internal bleeding], and blood clots.^1,2,3

The vaccine enters the platelet. The platelet makes the spike protein. The immune system attacks the infected platelets. Fewer platelets = bleeding, clots.

References
1. Might Post-Injection Distribution Of Covid Vaccines To The Brain Explain The Rare Fatal Events Of Cerebral Venous Sinus Thrombosis (CVST)?, British Medical Journal, 14 April 2021
2. CoViD Vaccines and thrombotic events: Possibility of mRNA translation and spike protein synthesis by platelets?, British Medical Journal, 11 March 2021
3. Re: COVID vaccines and thrombotic events: is mRNA translation and spike protein synthesis by platelets a real possibility?, British Medical Journal, 11 March 2021