This has been a controversial dispute for a long time and it can involve a lot of personal opinion, but I will try to answer this as scientifically as possible.
There hasn't been any viable evidence that vaccines do cause autism. Several different theories have been proposed on why vaccines could cause autism, such as the ingredient in some vaccines thimerosal being harmful, but these have all been disproved by many different experiments.
Many reliable sources such as the CDC1 says that there is no link between autism an vaccines. A 2011 report2 from the Institute of Medicine that tested 8 different vaccines on both adults and children showed that the vaccines are very safe.
A 2013 CDC study3 added to the research showing that vaccines do not cause ASD. The study looked at the number of antigens (substances in vaccines that cause the body’s immune system to produce disease-fighting antibodies) from vaccines during the first two years of life. The results showed that the total amount of antigen from vaccines received was the same between children with ASD and those that did not have ASD.
The CDC has also funded many other studies on vaccine ingredients and their links to autism, especially thimerosal4, a mercury-based ingredient used in some vaccines. These studies have shown "no link between thimerosal-containing vaccines and ASD, as well as no link between the measles, mumps, and rubella (MMR) vaccine and ASD in children."
Full list of CDC studies on the link between vaccines and autism5
These are pretty convincing, but you can't just rely on one source, right?
An issue in the Oxford Journal6 also shows how even though there have been many claims that vaccines do cause autism, there aren't any studies that help support this claim.
A worldwide increase in the rate of autism diagnoses—likely driven by broadened diagnostic criteria and increased awareness—has fueled concerns that an environmental exposure like vaccines might cause autism. Theories for this putative association have centered on the measles-mumps-rubella (MMR) vaccine, thimerosal, and the large number of vaccines currently administered. However, both epidemiological and biological studies fail to support these claims.
One study that took place in the UK from 1979-19927 found that even though there was an increase in cases of autism, there was no jump in the amount of cases when the measles-mumps-rubella (MMR) vaccine, another controversial vaccine, was introduced in 1988.
A 2003 study in metropolitan Atlanta8, Georgia compared the ages that the MMR vaccine was given in children with autism against the age that children without autism received it. The results were that the proportion of time that children were vaccinated in both control groups was similar.
The Oxford Journal entry mentioned many other studies that tested the link between MMR vaccines and autism and there has been no plausible evidence.
The other vaccine ingredient that is also said to cause autism is thimerosal, which I mentioned before. The Oxford Journal issue also shows studies about this link.
A study in Finland9 helped show that there wasn't a link between thimerosal and autism. In fact, cases of autism increased in 1992 after thimerosal-containing vaccines were discontinued.
Another study in the UK10 also disproved the theory that thimerosal causes autism. They actually found that thimerosal-containing vaccines might even have a beneficial effect in children.
There have also been many other studies that invalidate the link between thimerosal and autism.
One last theory that has been proposed is that multiple vaccines being administered to a child simultaneously can weaken their immune system and cause autism. This theory is flawed for several reasons.
This article11 used data from many studies and found that vaccines do not overwhelm and weaken an infant's immune system. Infants are able to respond well to the many vaccines that are given to them.
Another study12 found that vaccines did not weaken the immune system to any diseases. They were unable to find consistent relationship between infectious diseases and immunization.
All of this scientific evidence leaves one question left to be answered.
Why do any people believe that vaccines cause autism? (this part will get some more opinion then you may like, but it is necessary)
A fraudulent 1998 study13 by Andrew Wakefield and colleagues was one of the main reasons that this controversy over vaccines and autism began. It used falsified data14 to support their claim that the MMR vaccine caused developmental problems and other autism symptoms. This study has since been retracted as it was demonstrated to be false. Dr. Wakefield, a physician in the UK, was struck off the Medical Registry due to his participation in this fraud.
WebMD15 also talks about why there are still some people that believe that vaccines are the cause of autism.
And when something bad happens to a child, people demand to know what or whom is to blame. "Parents are clamoring for a cause," says David Tayloe, MD, a pediatrician in Greensboro, N.C., and president-elect of the American Academy of Pediatrics (AAP).
"It's a terrible condition. It upsets families, and it upsets me." But all the fear and anger about vaccines is misplaced, he says. "There's just nothing there."
It seems most people who claim that vaccines cause autism are just looking for something to blame for what is happening to children. Many parents may become angry when they find out their child has autism, and they put that anger against vaccines.
There are some logical arguments that people may present, such as changing of their child's behavior after receiving vaccines. This is mostly coincidence, as most children are vaccinated around the same time autism symptoms begin to appear (usually 12-18 months)16. Other than this and some of the arguments mentioned earlier, there is almost no reason to believe that vaccines cause autism.
 CDC - Vaccines Do Not Cause Autism
 Adverse Effects of Vaccines: Evidence and Causality
 Increasing Exposure to Antibody-Stimulating Proteins and Polysaccharides in Vaccines Is Not Associated with Risk of Autism
 CDC - Frequently Asked Questions About Thimerosal
 CDC - Vaccines and Autism: A Summary of CDC Conducted or Sponsored Studies
 Vaccines and Autism: A Tale of Shifting Hypotheses
 Autism and measles, mumps, and rubella vaccine: no epidemiological evidence for a causal association
 Age at First Measles-Mumps-Rubella Vaccination in Children With Autism and School-Matched Control Subjects: A Population-Based Study in Metropolitan Atlanta
 Thimerosal and the Occurrence of Autism: Negative Ecological Evidence From Danish Population-Based Data
 Thimerosal Exposure in Infants and Developmental Disorders: A Prospective Cohort Study in the United Kingdom Does Not Support a Causal Association
 Addressing Parents’ Concerns: Do Multiple Vaccines Overwhelm or Weaken the Infant’s Immune System?
 DTP Immunization and Susceptibility to Infectious Diseases
 RETRACTED: Ileal-lymphoid-nodular hyperplasia, non-specific colitis, and pervasive developmental disorder in children
 The MMR vaccine and autism: Sensation, refutation, retraction, and fraud
 WebMD - Autism-Vaccine Link: Evidence Doesn't Dispel Doubts
 When do children usually show symptoms of autism spectrum disorder (ASD)?