The Scotland national team qualified for the rescheduled European Championships in November, but there are concerns Scottish fans may be excluded from the crunch tie on June 18.
Under current Covid-19 measures, fans remain unable to attend stadiums in Scotland.
But the Health Secretary said it is up to the compliance of Scots to ensure fans can travel to Wembley.
Speaking at the coronavirus briefing in Edinburgh on Wednesday, Ms Freeman said: “Part of who holds the answer to that question is actually all of us.
“It’s all of us abiding by the rules, doing everything we can to make sure that the case numbers come down, the virus is suppressed.
“We try very hard, every step along the way toward that date in the summer, that we’re trying to do everything we can to make sure that the virus has no opportunity to transmit from one person to another and when we are, as adults, called for that vaccination appointment we take that up.
“Our intention is, supplies allowing, that we will have vaccinated every adult over the age of 18 in Scotland by the end of May.
“That is a really important and hopeful target to aim for, so if we can bring all of that together and the data continues to point us in the right direction, then we certainly should be able to celebrate our national team playing in the Euros – hopefully winning in the Euros – and we’ll see whether or not it’s possible at any point along that road for fans to actually be present to witness that.”
On Wednesday, Scotland recorded 35 deaths from coronavirus and 543 positive tests.
It brings the death toll under this measure – of people who first tested positive for the virus within the previous 28 days – to 7,371.
A total of 1,661,879 people have received their first dose of a coronavirus vaccine as of Wednesday morning and 8,105 have received their second dose.